Question

If g(x) is the inverse of f(x) and f(x) = 4x+12 what is g(x)?

Answer 1

Answer:

Inverse function: A function g is the inverse of a function f if whenever y=f(x) then x=g(y).

In other words, we can write this in terms of the composition of f and g as g(f(x))=x.

For any input x, the function corresponding to f spits out the value y=f(x)=4x+12.

Now, we want to find the inverse function g(x)=$f^{-1}$ that takes the value y as an input and spits out x as the output.

In other words, y=f(x) gives y as a function of x and we want to find $x=f^{-1}(y)$ that will give us x as a function of y.

Given,the expression y=4x+12 for y as a function of x and solve for x.

Subtract 12 from both sides we get;

y-12 = 4x+12-12

Simplify:

y-12 = 4x

Divide by 4 to both sides we get;

$\frac{y-12}{4} =\frac{4x}{4}$

Simplify:

$x=\frac{1}{4}y - 3$

therefore,  $x = f^{-1}(y) = \frac{1}{4}y-3$

since, g(x) is the inverse of f(x)

⇒$g(x)=\frac{1}{4}x-3$

Now, verify that g(x) is really the inverse of f(x), we should show that the composition of f and g doesn't do anything to the input.

$(g o f)(x) = g(f(x)) = g(4x+12) = \frac{1}{4}(4x+12) -3 = x+3 -3$

Simplify:

g(f(x)) = x                for all x

⇒  g(x) is the inverse of f(x)

Therefore, $g(x)=\frac{1}{4}x-3$

Answer 2

Using inverse functions, it is found that that:

$g(x) = \frac{x - 12}{4}$

To find the inverse function, we exchange x and y in the original function, then isolate f.

The function f(x) is given by:

$f(x) = 4x + 12$

Function g(x) is the inverse of f(x), then:

$y = 4x + 12$

$x = 4y + 12$

$4y = x - 12$

$y = \frac{x - 12}{4}$

$g(x) = \frac{x - 12}{4}$

To learn more about inverse functions, you can take a look at https://brainly.com/question/16485117