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Question

A function is given:

f(x)=3x+12

Determine the inverse of this function and name it g(x).

Use composite functions to show that these functions are inverses.
f(g(–2)).
Explain: What is the domain?

Answer 1

To find the inverse of the given function f(x) = 3x + 12, we need to switch the roles of x and y and solve for y.

So, start with:
y = 3x + 12
Switch x and y:
x = 3y + 12
Solve for y:
x - 12 = 3y
y = (x - 12) / 3

Therefore, the inverse function, g(x), is g(x) = (x - 12) / 3.

To show that f(x) and g(x) are inverses, we need to show that f(g(x)) = x and g(f(x)) = x.

Let's start with f(g(-2)):
f(g(-2)) = f((-2 - 12) / 3)
f(g(-2)) = f(-14 / 3)
f(g(-2)) = 3(-14 / 3) + 12
f(g(-2)) = -14 + 12
f(g(-2)) = -2

Now, let's check the other composite function:
g(f(x)):
g(f(x)) = g(3x + 12)
g(f(x)) = ((3x + 12) - 12) / 3
g(f(x)) = 3x / 3
g(f(x)) = x

Since both composite functions simplify to x, we have shown that f(x) and g(x) are inverses of each other.

The domain of the original function f(x) = 3x + 12 is all real numbers since there are no restrictions on the input x.