Find f(g(x)), g(f(x))

Question

Consider the functions f(x)= -4x+9 and g(x)= -1/4(x-9)

(a) Find f(g(x))

(b) Find g(f(x))

(c) Determine whether the functions f and g are inverses of each other.

Answer 1

When f(x) = -4x + 9, then f(1) is found by replacing all terms in f(x) with the variable x with the value 1 and thus we would get -4 (1) + 9 = 5. For g(x) = -1/4 (x - 9), it follows that f(g(x)) is found by replacing all terms in f(x) with the variable x with g(x), so we have f(g(x) = - 4 [-1/4 * (x-9)] + 9 = x

likewise g(f(x)) is found by replacing all values of x in g(x) with the function f(x)

-1/4 *([-4x + 9] - 9) = x

Finally go show that f(x) and g(x) are inverses of each other, then f(g(x)) and g(f(x)) both need to = x, which they do so f and g are inverses of each other

Answer 2

Answers: a) x b) x c) Yes

(a) f(g(x))

Take the function of f(x) and plug in g(x) for x.

-4x+9

-4(-1/4(x-9))+9 Follow the order of operations, start with parentheses

-4(-1/4x+9/4)+9 Distribute -4

x-9+9

x

(b) g(f(x))

Take the function of g(x) and plug in f(x) for x.

-1/4(x-9)

-1/4(-4x+9-9) Combine like terms inside parantheses

-1/4(-4x) Multiply

x

(c) The functions are inverse if they have the same value plugged into each other. Since they both equaled x, they are inverses of each other.