Exploring Composite Functions with Fractions and Radicals

This video tutorial covers composite functions, including how to find them, determine their domains, and use them to test if two functions are inverses. It provides examples of composite functions, explores domain restrictions, and introduces inverse functions with graphical representations. The video also demonstrates methods to prove that two functions are inverses of each other.

What is a composite function?

• A function that is added to another function.

• A function that is multiplied by another function.

• A function inside another function.

• A function that is divided by another function.

In the composite function f(g(x)), which function is the input?

• Neither f(x) nor g(x)

• f(x)

• Both f(x) and g(x)

• g(x)

What is the domain of the composite function f(g(x)) if g(x) = x + 3 and f(x) = 1/x?

• All real numbers except 3

• All real numbers except 0

• All real numbers except -3

• All real numbers

How do you represent the domain of a function using interval notation?

• Using curly braces for all numbers

• Using square brackets for included numbers and parentheses for excluded numbers

• Using parentheses for all numbers

• Using square brackets for all numbers

What is the domain of the composite function g(f(x)) if f(x) = 1/x and g(x) = x + 3?

• All real numbers

• All real numbers except 3

• All real numbers except -3

• All real numbers except 0

What is the first step in finding the inverse of a function f(x)?

• Switch x and y

• Replace f(x) with y

• Solve for y

• Graph the function

What is the graphical relationship between a function and its inverse?

• They are reflections about the line y = x

• They intersect at the origin

• They are perpendicular to each other

• They are parallel to each other

How can you prove that two functions are inverses of each other?

• By checking their domains

• By solving for y in both functions

• By finding their composite functions and showing both equal x

• By showing their graphs intersect

If f(x) = 3x + 5, what is its inverse function?

• x + 5 / 3

• 3 / (x - 5)

• x - 5 / 3

• 3x - 5

Are the functions f(x) = x^3 + 3 and g(x) = cube root(x - 3) inverses of each other?

• Yes

• No

• Only if x is positive

• Only if x is negative