Question

Which of the is the inverse of (\f(x) = 4x+12\)?

Answer 1

Find the Inverse Function f(x)=4x-12

f(x)=4x−12f(x)=4x-12

Replace f(x)f(x) with yy.

y=4x−12y=4x-12

Interchange the variables.

x=4y−12x=4y-12

Solve for yy.



Since yy is on the right side of the equation, switch the sides so it is on the left side of the equation.

4y−12=x4y-12=x

Add 1212 to both sides of the equation.

4y=12+x4y=12+x

Divide each term by 44 and simplify.



Divide each term in 4y=12+x4y=12+xby 44.

4y4=124+x44y4=124+x4

Reduce the expression by cancelling the common factors.



Cancel the common factor.

4y4=124+x44y4=124+x4

Divide yy by 11.

y=124+x4y=124+x4

Divide 1212 by 44.

y=3+x4y=3+x4

Solve for yy and replace with f−1(x)f-1(x).



Replace the yy with f−1(x)f-1(x)to show the final answer.

f−1(x)=3+x4f-1(x)=3+x4

Set up the composite result function.

f(g(x))f(g(x))

Evaluate f(g(x))f(g(x)) by substituting in the value of gg into ff.

4(3+x4)−124(3+x4)-12

Simplify each term.



Apply the distributive property.

f(3+x4)=4⋅3+4(x4)−12f(3+x4)=4⋅3+4(x4)-12

Multiply 44 by 33.

f(3+x4)=12+4(x4)−12f(3+x4)=12+4(x4)-12

Cancel the common factor of 44.

Write 44 as a fraction with denominator 11.

f(3+x4)=12+41⋅x4−12f(3+x4)=12+41⋅x4-12

Factor out the greatest common factor 44.

f(3+x4)=12+4⋅11⋅x4⋅1−12f(3+x4)=12+4⋅11⋅x4⋅1-12

Cancel the common factor.

f(3+x4)=12+4⋅11⋅x4⋅1−12f(3+x4)=12+4⋅11⋅x4⋅1-12

Rewrite the expression.

f(3+x4)=12+11⋅x1−12f(3+x4)=12+11⋅x1-12

Simplify.


Multiply 1111 and x1x1.

f(3+x4)=12+x1−12f(3+x4)=12+x1-12

Divide xx by 11.

f(3+x4)=12+x−12f(3+x4)=12+x-12

Simplify by subtracting numbers.


Subtract 1212 from 1212.

f(3+x4)=x+0f(3+x4)=x+0

Add xx and 00.

f(3+x4)=xf(3+x4)=x

Since f(g(x))=xf(g(x))=x, f−1(x)=3+x4f-1(x)=3+x4 is the inverse of f(x)=4x−12f(x)=4x-12.

f−1(x)=3+x4

Answer 2

The inverse of f(x) = 4x + 12 is given by

$\displaystyle \sf{ } {f}^{- 1} (x) = \frac{x - 12}{4}$

Given : The given function f(x) = 4x + 12

To find : The inverse of the function

Solution :

Here the given function is

f(x) = 4x + 12

$\sf Let \: \: {f}^{- 1} (x) = y$

$\displaystyle \sf{ \implies f(y) = x}$

$\displaystyle \sf{ \implies 4y + 12= x}$

$\displaystyle \sf{ \implies 4y = x - 12}$

$\displaystyle \sf{ \implies y = \frac{x - 12}{4} }$

$\displaystyle \sf{ \implies } {f}^{- 1} (x) = \frac{x - 12}{4}$