Inverse Functions & Composition

Front

What is an inverse function?

Back

An inverse function reverses the operation of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x.

Tags

CCSS.HSF-BF.B.4A

Front

How do you find the inverse of a linear function?

Back

To find the inverse of a linear function f(x) = ax + b, swap x and y, then solve for y. The result is f^-1(x) = (x - b)/a.

Tags

CCSS.HSF-BF.B.4A

Front

What is the composition of functions?

Back

The composition of functions f and g, denoted as (f⋅g)(x), means applying g first and then applying f to the result: (f⋅g)(x) = f(g(x)).

Tags

CCSS.HSF-BF.A.1C

Front

Find the inverse of f(x) = 1/4x - 7.

Back

f^-1(x) = 4x + 28.

Tags

CCSS.HSF-BF.B.4A

Front

If f(x) = 4x + 8 and g(x) = x + 3, what is (f⋅g)(x)?

Back

(f⋅g)(x) = 4x² + 20x + 24.

Tags

CCSS.HSF-BF.A.1C

Front

Calculate f(g(5)) for f(x) = 3x + 10 and g(x) = x - 2.

Back

f(g(5)) = 19.

Tags

CCSS.HSF-BF.A.1C

Front

What is the formula for the inverse of a quadratic function?

Back

Quadratic functions do not have inverses unless restricted to a domain where they are one-to-one. For example, f(x) = ax² + bx + c can be inverted using the quadratic formula.

Tags

CCSS.HSF-BF.B.4D

Front

Find g(f(-10)) for f(x) = 3x + 10 and g(x) = x - 2.

Back

g(f(-10)) = None of these.

Tags

CCSS.HSF-BF.A.1C

Front

What is the significance of the horizontal line test?

Back

The horizontal line test determines if a function has an inverse. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse.

Front

How do you verify if two functions are inverses?

Back

To verify if f and g are inverses, check if f(g(x)) = x and g(f(x)) = x for all x in the domain.

Tags

CCSS.HSF-BF.B.4B