Finding inverse functions

Learn how to find the formula of the inverse function of a given function. For example, find the inverse of f(x)=3x+2.
Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if ff takes aa to bb, then the inverse, f1f^{-1}, must take bb to aa.
Or in other words, f(a)=bf1(b)=a f(a)=b \iff f^{-1}(b)=a.
In this article we will learn how to find the formula of the inverse function when we have the formula of the original function.

Before we start...

In this lesson, we will find the inverse function of f(x)=3x+2f(x)=3x+2.
Before we do that, let's first think about how we would find f1(8)f^{-1}(8).
To find f1(8)f^{-1}(8), we need to find the input of ff that corresponds to an output of 88. This is because if f1(8)=xf^{-1}(8)=x, then by definition of inverses, f(x)=8f(x)=8.
f(x)=3x+28=3x+2Let f(x)=86=3xSubtract 2 from both sides2=xDivide both sides by 3\begin{aligned} f(x) &= 3 x+2\\\\ 8 &= 3 x+2 &&\small{\gray{\text{Let f(x)=8}}} \\\\6&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ 2&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
So f(2)=8f(2)=8 which means that f1(8)=2f^{-1}(8)=2

Finding inverse functions

We can generalize what we did above to find f1(y)f^{-1}(y) for any yy.
Note that while the choice of variable is arbitrary, we must choose something different than xx as this is already involved in the function.
We can change it back to xx in the end if desired.
To find f1(y)f^{-1}(y), we can find the input of ff that corresponds to an output of yy. This is because if f1(y)=xf^{-1}(y)=x then by definition of inverses, f(x)=yf(x)=y.
f(x)=3x+2y=3x+2Let f(x)=yy2=3xSubtract 2 from both sidesy23=xDivide both sides by 3\begin{aligned} f(x) &= 3 x+2\\\\ y &= 3 x+2 &&\small{\gray{\text{Let f(x)=y}}} \\\\y-2&=3x &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ \dfrac{y-2}{3}&=x &&\small{\gray{\text{Divide both sides by 3}}} \end{aligned}
So f1(y)=y23f^{-1}(y)=\dfrac{y-2}{3}.
Since the choice of the variable is arbitrary, we can write this as f1(x)=x23f^{-1}(x)=\dfrac{x-2}{3}.
Because inverse functions reverse the inputs and outputs, another way to find f1f^{-1} is by switching xx and yy initially, then solving for yy to write the inverse in function form.
f(x)=3x+2y=3x+2Replace f(x) with yx=3y+2Switch x and yx2=3ySubtract 2 from both sidesx23=yDivide both sides by 3\begin{aligned} f(x)&=3x+2\\\\ \goldD y &= 3\blueD x+2 &&\small{\gray{\text{Replace f(x) with y}}}\\\\ \blueD x &= 3\goldD y+2 &&\small{\gray{\text{Switch x and y}}} \\\\x-2&=3y &&\small{\gray{\text{Subtract 2 from both sides}}}\\\\ \dfrac{x-2}{3}&=y &&\small{\gray{\text{Divide both sides by 3}}} \\\\ \end{aligned}
So f1(x)=x23f^{-1}(x)=\dfrac{x-2}{3}.
Notice the similarities among the two methods. In each case, the equation is solved for the variable that is not isolated. The main difference involves when the variables are reversed. (The first method does this at the end while the second method does this initially.)

Check your understanding

1) Linear function

Find the inverse of g(x)=2x5g(x)=2x-5.
g1(x)=g^{-1}(x)=

To start, replace g(x)g(x) with yy. Then solve for xx.
g(x)=2x5y=2x5Replace g(x) with yy+5=2xAdd 5 to both sidesy+52=xDivide both sides by 2y+52=g1(y)Replace x with g1(y)\begin{aligned} g(x)&= 2x-5 \\\\ y &= 2x-5 &&\small{\gray{\text{Replace g(x) with y}}}\\\\ y+5 &= 2x &&\small{\gray{\text{Add 5 to both sides}}}\\\\ \dfrac{y+5}{2}&=x &&\small{\gray{\text{Divide both sides by 2}}} \\\\ \dfrac{y+5}{2}&=g^{-1}(y) &&\small{\gray{\text{Replace x with }g^{-1}(y)}} \end{aligned}
Since the choice of variable is arbitrary, we can now switch yy to xx to write the inverse in terms of xx.
g1(x)=x+52g^{-1}(x)=\dfrac{x+5}{2}

2) Cubic function

Find the inverse of h(x)=x3+2h(x)=x^3+2.
h1(x)=h^{-1}(x)=

To start, replace h(x)h(x) with yy. Then solve for xx.
Since the choice of variable is arbitrary, we can now switch yy to xx to write the inverse in terms of xx.
h1(x)=x23h^{-1}(x)=\sqrt[3]{x-2}

3) Cube-root function

Find the inverse of f(x)=4x3f(x)=4\cdot \sqrt[\Large3]{x}.
f1(x)=f^{-1}(x)=

To start, replace f(x)f(x) with yy. Then solve for xx.
f(x)=4x3y=4x3Replace f(x) with yy4=x3Divide both sides by 4(y4)3=xCube both sides(y4)3=f1(y)Replace x with f1(y)\begin{aligned} f(x)&= 4\sqrt[3]{x} \\\\ y &= 4\sqrt[3]{x} &&\small{\gray{\text{Replace f(x) with y}}}\\\\ \dfrac{y}{4} &=\sqrt[3]{x}&&\small{\gray{\text{Divide both sides by 4}}}\\\\ \left(\dfrac{y}{4}\right)^3&=x&&\small{\gray{\text{Cube both sides}}} \\\\ \left(\dfrac{y}{4}\right)^3&=f^{-1}(y)&&\small{\gray{\text{Replace x with }f^{-1}(y)}} \\\\ \end{aligned}
Since the choice of variable is arbitrary, we can now switch yy to xx to write the inverse in terms of xx.
So f1(x)=(x4)3f^{-1}(x)=\left(\dfrac{x}{4}\right)^3 or f1(x)=x364f^{-1}(x)=\dfrac{x^3}{64}

4) Rational functions

Find the inverse of g(x)=x3x2g(x)=\dfrac{x-3}{x-2}.
g1(x)=g^{-1}(x)=

To start, replace g(x)g(x) with yy. Then solve for xx.
g(x)=x3x2y=x3x2Replace g(x) with yy(x2)=x3Multiply both sides by x-2yx2y=x3Distributeyxx=2y3Group terms with x on one sidex(y1)=2y3Factor out an xx=2y3y1Divide both sides by y-1g1(y)=2y3y1Replace x with g1(y)\begin{aligned} g(x)&= \dfrac{x-3}{x-2} \\\\\\ y&= \dfrac{x-3}{x-2} &&\small{\gray{\text{Replace g(x) with y}}}\\\\\\ y(x-2) &= x-3 &&\small{\gray{\text{Multiply both sides by x-2}}}\\\\ yx-2y&= x-3 &&\small{\gray{\text{Distribute}}} \\\\ yx-x&=2y-3&&\small{\gray{\text{Group terms with x on one side}}}\\\\ x(y-1)&=2y-3&&\small{\gray{\text{Factor out an x}}}\\\\ x&=\dfrac{2y-3}{y-1} &&\small{\gray{\text{Divide both sides by y-1}}}\\\\ g^{-1}(y)&=\dfrac{2y-3}{y-1} &&\small{\gray{\text{Replace x with }g^{-1}(y)}}\end{aligned}
Since the choice of variable is arbitrary, we can now switch yy to xx to write the inverse in terms of xx.
g1(x)=2x3x1g^{-1}(x)=\dfrac{2x-3}{x-1}

5) Challenge problem

Match each function with the type of its inverse.
Function
Inverse type
  • f(x)=3x5f(x)=3x-5
  • g(x)=x37g(x)=x^3-7
  • h(x)=x+2x3h(x)=\dfrac{x+2}{x-3}
  • j(x)=x+2j(x)=\sqrt{x+2}
  • Quadratic
  • Cube root
  • Linear
  • Rational

You can complete this problem without finding the actual inverses. Just think about what operations would "undo" the function. Or, use what was done above to help you.
FunctionInverse type
f(x)=3x5f(x)=3x-5Linear
g(x)=x37g(x)=x^3-7Cube root
h(x)=x+2x3h(x)=\dfrac{x+2}{x-3}Rational
j(x)=x+2j(x)=\sqrt{x+2}Quadratic
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