How to Find the Domain of Any Function (NancyPi) @NancyPi

NancyPi | How to Find the Domain of Any Function (NancyPi) @NancyPi | Uploaded May 2018 | Updated October 2024, 3 hours ago.
MIT grad shows a surefire way to find the domain of any function. To skip ahead: 1) For POLYNOMIAL only, skip to time 0:45. 2) For FRACTION only, skip to time 1:44. 3) For SQUARE ROOT only, skip to 5:08. 4) For SQUARE ROOT ON BOTTOM of Fraction, skip to 8:44. 5) For SQUARE ROOT ON TOP of Fraction, skip to 10:09. Nancy formerly of MathBFF explains the steps.

For how to SOLVE QUADRATIC equations by FACTORING, jump to: youtu.be/Z5MnP9da4EM
For how to SOLVE INEQUALITIES, jump to: youtu.be/ONuIIvgkE3c

Follow Nancy on Instagram! instagram.com/nancypi
Twitter: twitter.com/nancypi

For Domain, you just need to watch out for Square Roots and/or Fractions!

If your function is...

1) a POLYNOMIAL (no square roots or fractions), i.e. f(x) = x^2 + 3x + 1, domain is "all real numbers."

2) a FRACTION (w/ no square root), i.e. f(x) = (2x+1)/(x^2+5x+6). Set bottom "not equal" to zero. Solve for x.

3) a SQUARE ROOT (w/ no fraction), i.e. f(x) = sqrt(x+1). Set part under root "greater than or equal to zero".

4) a FRACTION w/ A SQUARE ROOT ON BOTTOM, i.e. f(x) = (x^2+2x+3)/(sqrt(x+1)). Set part under root "greater than zero." Solve the inequality. NOTE THE DIFFERENCE: You must set it greater than zero, not "greater than OR EQUAL TO zero" because we can't have zero (or square root of zero) in the denominator.

5) a FRACTION w/ A SQUARE ROOT ON TOP, i.e. f(x) = (sqrt(x+1))/(x^2+4), i) Set the bottom "not equal to" zero & solve; ii) Set what's under the root "greater than or equal to zero" & solve; iii) Take intersection of your two solutions.

For more of my math videos, check out: nancypi.com

$How to Find the Domain of Any Function (NancyPi) MIT grad shows a surefire way to find the domain of any function. To skip ahead: 1) For POLYNOMIAL only, skip to time 0:45. 2) For FRACTION only, skip to time 1:44. 3) For SQUARE ROOT only, skip to 5:08. 4) For SQUARE ROOT ON BOTTOM of Fraction, skip to 8:44. 5) For SQUARE ROOT ON TOP of Fraction, skip to 10:09. Nancy formerly of MathBFF explains the steps. For how to SOLVE QUADRATIC equations by FACTORING, jump to: https://youtu.be/Z5MnP9da4EM For how to SOLVE INEQUALITIES, jump to: https://youtu.be/ONuIIvgkE3c Follow Nancy on Instagram! https://instagram.com/nancypi Twitter: https://twitter.com/nancypi For Domain, you just need to watch out for Square Roots and/or Fractions! If your function is... 1) a POLYNOMIAL (no square roots or fractions), i.e. f(x) = x^2 + 3x + 1, domain is all real numbers. 2) a FRACTION (w/ no square root), i.e. f(x) = (2x+1)/(x^2+5x+6). Set bottom not equal to zero. Solve for x. 3) a SQUARE ROOT (w/ no fraction), i.e. f(x) = sqrt(x+1). Set part under root greater than or equal to zero. 4) a FRACTION w/ A SQUARE ROOT ON BOTTOM, i.e. f(x) = (x^2+2x+3)/(sqrt(x+1)). Set part under root greater than zero. Solve the inequality. NOTE THE DIFFERENCE: You must set it greater than zero, not greater than OR EQUAL TO zero because we cant have zero (or square root of zero) in the denominator. 5) a FRACTION w/ A SQUARE ROOT ON TOP, i.e. f(x) = (sqrt(x+1))/(x^2+4), i) Set the bottom not equal to zero & solve; ii) Set whats under the root greater than or equal to zero & solve; iii) Take intersection of your two solutions. For more of my math videos, check out: http://nancypi.com$

$How to Solve Inequalities (NancyPi) MIT grad explains solving inequalities. This video focuses on solving linear inequalities. It shows when to switch the sign of the inequality, if you divide or multiply by a negative number, and is an introduction to how to solve inequalities in algebra. To skip ahead: 1) For a basic example of SOLVING AN INEQUALITY, skip to time 00:34. 2) For when you have to FLIP THE INEQUALITY SIGN when DIVIDING BY A NEGATIVE number, skip to 1:26. 3) For when you need to reverse the inequality symbol because youre MULTIPLYING BY A NEGATIVE number, skip to 2:42. The problems here are multi-step inequalities, or two-step inequalities. Nancy formerly of MathBFF explains the steps. Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi Algebraic equations and inequalities: what is inequality math? What does inequality mean? An inequality is a number sentence that uses an inequality symbol instead of an equality symbol (=), which means it has a greater than sign, less than sign, greater than or equal to sign, or less than or equal to sign, instead of the normal equal sign in an equation. One linear inequality definition is: an inequality that involves a linear algebraic expression and inequality signs. How to find the solution of an inequality algebra problem: 1) A simple, BASIC LINEAR INEQUALITY example you might see on a solving inequalities worksheet or problem set is the expression 2x + 3 [less than or equal to] 11. You can solve for x just like you would for an equality, or normal equation, by trying to get x alone. This is an example of two-step inequalities, or multi-step inequalities. The first step is to subtract the constant 3 from both sides, in order to move it to the right hand side, so that you have 2x [less than or equal to] 8. The second step is to get x alone on the left side by dividing both sides by 2 so that your answer for the inequality is x [less than or equal to] 4. In this case, solving the inequality used the exact same steps you would use to solve an equal sign equation. Sometimes the steps are different from solving normal math equations. Solving for x is the same as solving a linear equality (-2x + 3 = 11) except that if you ever have to divide by a negative number or multiply by a negative number while solving for x, you must flip the sign, or switch the direction of your inequality symbol less than becomes greater than, for example). Here are the two cases of inequality problems WHEN YOU NEED TO REVERSE THE SIGN of the inequality because youre multiplying or dividing both sides of the inequality by a negative number: 2) DIVIDING EXAMPLE: For an inequality like -2x + 3 [less than or equal to] 11, after moving the constant 3 to the right side, you will need to divide both sides by negative 2 in order to get x alone on the left side for the solution. WHEN YOU DIVIDE BY A NEGATIVE NUMBER, you need to reverse the direction of the inequality symbol in your expression. In this example, after you divide out -2, you get x [greater than or equal to] -4 (instead of x less than or equal to -4) for the final solution. This example was also a two-step inequality problem. 3) MULTIPLYING EXAMPLE: For examples like (-1/2)x + 4 [greater than] 7, when youve moved the constant to the right side and have the inequality (-1/2)x [greater than] 3, the fastest way to get x alone is to clear the fraction by multiplying both sides by -2. This will cancel the 2 in the denominator and remove the negative sign in front of x on the left side. But WHEN YOU MULTIPLY BY A NEGATIVE, you have to flip the direction of the inequality symbol so that your answer is x [less than] -6 instead of x [greater than] -6. This problem was also a multi-step inequality. Note on inequality rules: you dont have to switch the symbol if youre adding a negative number or subtracting a negative number: switching the inequality sign is only for when youre multiplying or dividing by a negative number. For quadratic inequalities examples, compound inequalities, inequality math, inequality algebra, inequality equations, inequality symbols, inequality notation, and more algebra solver, equation solver, and inequality calculator practice problems that come up on worksheets, as well as more algebra, algebra 2, and college algebra mathematics videos, check out: http://nancypi.com$

$Derivatives... How? (NancyPi) MIT grad shows how to find derivatives using the rules (Power Rule, Product Rule, Quotient Rule, etc.). To skip ahead: 1) For how and when to use the POWER RULE, constant multiple rule, constant rule, and sum and difference rule, skip to time 0:22. 2) For the PRODUCT RULE, skip to 7:36. 3) For the QUOTIENT RULE, skip to 10:53. For my video on the CHAIN RULE for finding derivatives: https://youtu.be/H-ybCx8gt-8 For my video on the DEFINITION of the derivative: https://youtu.be/-ktrtzYVk_I Nancy formerly of MathBFF explains the steps. For more of the QUOTIENT RULE and a shortcut to remember the formula, jump to my video at: https://youtu.be/jwuiVb84Xx4 For Follow Nancy on Instagram: https://instagram.com/nancypi Twitter: https://twitter.com/nancypi What is the derivative? Its a function that gives you the instantaneous rate of change at each point of another function. You can calculate the derivative with the definition of the derivative (using the limit), but the fastest way to find the derivative is with shortcuts such as the Power Rule, Product Rule, and Quotient Rule: 1) POWER RULE: If the given equation is a polynomial, or just a power of x, then you can use the Power Rule. For a term thats just a power of x, such as x^4, you can get the derivative by bringing down the power to the front of the term as a coefficient and decreasing the x power by 1. For example, for x^4, the derivative is 4x^3. If you have many terms added or subtracted together, and if they are powers of x, you can use the Power Rule on each term (by the Sum and Difference Rules). NOTE: The derivative of a constant, just a number, is always 0 (that is the Constant Rule). Also, if you have a term that is a constant multiplied in the front of the term, like 2x^3, you can keep the constant and differentiate the rest of the term. In this example, you keep the 2 and take the derivative of x^3, which is 3x^2, so the derivative of the term 2x^3 is 2*3x^2, or 6x^2. ANOTHER NOTE:You can use the same power rule method for fractional or negative powers, but be careful... for negative powers, it works as long as x is not 0, and for fractional/rational powers, if the power is less than 1, your derivative wont be defined at x = 0. 2) PRODUCT RULE: If your equation is not a polynomial but instead has the overall form of one expression multiplied by another expression, then you can use the Product Rule. The Product Rule says that the derivative of two functions multiplied together is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. 3) QUOTIENT RULE: If your equation has the overall form of one expression divided by another expression, then you can use the Quotient Rule. The Quotient Rule says that the derivative of one function divided by another (a quotient) is equal to the bottom function times the derivative of the top bottom minus the top function times the derivative of the bottom function, all divided by the bottom function squared. This is true as long as the bottom function is not equal to 0. For more of my math videos, check out: http://nancypi.com$

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$Power Rule... How? (NancyPi) MIT grad shows how to find the derivative using the Power Rule, one of the derivative rules in calculus. It is a shortcut for taking derivatives of polynomial functions with powers of x. To skip ahead: 1) For HOW and WHEN to use the power rule, skip to time 0:22. 2) For how to use the power rule when you have a FRACTIONAL or NEGATIVE POWER, skip to 5:22. For my video on the other differentiation rules, PRODUCT RULE and QUOTIENT RULE, skip to https://youtu.be/QqF3i1pnyzU?t=456 Nancy formerly of mathbff explains the steps: For how to differentiate using the formal LIMIT DEFINITION of the derivative instead, jump to https://youtu.be/-ktrtzYVk_I?t=628 Support Nancy on Patreon: https://www.patreon.com/nancypi Follow Nancy on Instagram: https://instagram.com/nancypi Follow Nancy on Twitter: https://twitter.com/nancypi HOW and WHEN to use the POWER RULE: If the given equation is a polynomial, or just a power of x, then you can use the Power Rule. For a term thats just a power of x, such as x^4, you can get the derivative by bringing down the power to the front of the term as a coefficient and decreasing the x power by 1. For example, for x^4, the derivative is 4x^3. If you have many terms added or subtracted together, and if they are powers of x, you can use the Power Rule on each term (by the Sum and Difference Rules). NOTE: The derivative of a constant, just a number, is always 0 (that is the Constant Rule). Also, if you have a term that is a constant multiplied in the front of the term, like 2x^3, you can keep the constant and differentiate the rest of the term (Constant Multiple Rule). In this example, you keep the 2 and take the derivative of x^3, which is 3x^2, so the derivative of the term 2x^3 is 2*3x^2, or 6x^2. ANOTHER NOTE: You can use the same power rule method for fractional or negative powers, but be careful... for negative powers, it works as long as x is not 0, and for fractional/rational powers, if the power is less than 1, your derivative wont be defined at x = 0. The derivative is a function that gives you the instantaneous rate of change at each point of another function. You can calculate the derivative with the definition of the derivative (using the limit, see https://youtu.be/-ktrtzYVk_I?t=628), but the fastest way to find the derivative is with shortcuts such as the Power Rule, Product Rule, and Quotient Rule. For my video on the CHAIN RULE for finding derivatives, jump to https://youtu.be/H-ybCx8gt-8 For more math help and videos, check out: http://nancypi.com Editor: Miriam Nielsen of zentouro https://www.youtube.com/zentouro Director: Kristopher Knight https://kristopherknight.com$ $More Chain Rule (NancyPi) MIT grad shows how to use the chain rule for EXPONENTIAL, LOG, and ROOT forms and how to use the chain rule with the PRODUCT RULE to find the derivative. To skip ahead: 4) For an example with an EXPONENTIAL function that needs the chain rule to take the derivative, skip to time 0:32. 5) For an example with a natural LOG, skip to 2:34. 6) For an example with a SQUARE ROOT, skip to 4:41. 7) For how to use the CHAIN RULE with the PRODUCT RULE, and how to know which to use first, skip to 7:17. For an introduction of HOW and WHEN to use the chain rule, jump to my FIRST chain rule video: https://youtu.be/H-ybCx8gt-8. Nancy formerly of MathBFF explains the steps. Support Nancy on Patreon: https://www.patreon.com/nancypi Follow Nancy on Instagram: https://instagram.com/nancypi Follow Nancy on Twitter: https://twitter.com/nancypi The CHAIN RULE is one of the derivative rules. You need it to take the derivative when you have a function inside a function, or a composite function. This video picks up at the end of the first chain rule video (Chain Rule...How? When? at https://youtu.be/H-ybCx8gt-8) and shows more examples of using the chain rule, as well as how to combine the chain rule with the product rule. For an intro of how to take derivatives, jump to: https://youtu.be/QqF3i1pnyzU 4) EXPONENTIAL: for the example y = e^(3x), first take the derivative of the outside function (the exponential function, e). The derivative of the exponential function is just the exponential function itself. When writing the outside derivative, leave the inside function (3x) the same, so write e^(3x). Then by the chain rule, you multiply by the derivative of just the inner function (3x). Since the inside derivative is 3, the answer for the full derivative dy/dx is 3e^(3x). 5) NATURAL LOG example: To find the derivative of y = ln(5x), use the same steps as above to first take the outside derivative and then multiply by the inside derivative. 5x is inside the natural log. The outer function here is the natural log, ln. To take the derivative of the natural log of something, we write 1 over that something, so 1 over 5x. We leave the inside function alone here when writing the outside derivative, so we write 1/(5x). Remember that were not done yet. We also need to multiply by the derivative of the inside function. The derivative of 5x is just 5, so our final simplified answer for dy/dx is 1/x. 6) SQUARE ROOT example: For this example, y = sqrt(x^2 + 1), the x^2 + 1 is under a square root. The square root is the outside, outer function. When you have a root of any kind, before you take the derivative, its easiest to rewrite it in the form of a fractional power, so that you can use the POWER RULE when taking its derivative. The outside function here is then the 1/2 power, and the inside function is x^2 + 1. To take the derivative of the outside one-half power, we use the power rule to bring down the 1/2 power to the front and reduce the old 1/2 power by 1. We then multiply by the inside derivative, and after simplifying get a derivative of y = x / sqrt(x^2 + 1). Here I used Lagrange notation (y) for the derivative instead of the Liebniz notation from before (dy/dx). To learn the POWER RULE, jump to my video at: https://youtu.be/TgIl15Nlg_U 7) CHAIN RULE WITH THE PRODUCT RULE: Sometimes you might need to use the chain rule combined with the product rule. How do you know whether to apply the chain rule first, or the product rule first? In this example, y = x^3 (2x - 5)^4, although the second factor here will need the chain rule for its derivative, overall in the biggest view of this function, we have a product of two factors. So we will use the product rule first here on the two factors, and then use the chain rule inside the product rule. However, for an example like ln[x^3 (2x-5)^4)], you would need to apply the chain rule first on the ln form, and then since your inside function is a product, you would need the product rule after that. Basically, the first thing youll do is whatever you need to use to handle the outermost form of your function. For more calculus math help and videos, check out: http://nancypi.com Editor: Miriam Nielsen of zentouro https://www.youtube.com/zentouro$

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