Finding the equation of a line from two points on its inverse

Finding an inverse function using coordinate points

Here we’ll look at how to translate from a linear function and its inverse when we’re given two points.

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The nice thing about functions and their inverses is that if you know two points, say $(a_1, b_1)$ and $(a_2, b_2)$, on a function $f(x)$, then two points on its inverse $f^{-1}(x)$ need to be $(b_1, a_1)$, and $(b_2, a_2)$. This works out very nicely if we know two points on a line and we want to find the inverse function.

Given two coordinate points on the inverse function, we can find the original function, assuming the functions are linear

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Two more examples starting with points on the inverse function

Example

Use the given information to find $f(x)$ if $f^{-1}(x)$ is a linear function.

$f^{-1}(3)=4$

$f^{-1}(-1)=5$

These are the points $(3,4)$ and $(-1,5)$ on the function $f^{-1}(x)$, which is the inverse of $f(x)$. That means on the function $f(x)$ the points will be $(4,3)$ and $(5,-1)$. Now we can use these points on the line $f(x)$ to find the equation of the line. Let’s begin by finding the slope $m$.

$m=\frac{3-(-1)}{4-5}=\frac{4}{-1}=-4$

Let’s find the $y$-intercept. We can use the slope we just found $m=-4$ and the equation of a line in slope intercept form, $y=mx+b$, along with a point to solve for $b$ (let’s use $(4,3)$).

$3=-4(4)+b$

$3=-16+b$

$3+16=b$

$19=b$

The equation of $f(x)$ is then

$f(x)=-4x+19$

This works out very nicely if we know two points on a line and we want to find the inverse function.

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If you like you can also use the points to find the function first and then find its inverse.

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Example

Use the given information to find $f(x)$ if $f^{-1}(x)$ is a linear function.

$f^{-1}(-2)=8$

$f^{-1}(-5)=14$

Let’s begin by finding the linear equation for $f^{-1}(x)$.

Use the points $(-2,8)$ and $(-5,14)$ to find the slope of the line $f^{-1}(x)$.

$m=\frac{14-8}{-5-(-2)}=\frac{6}{-3}=-2$

Let’s use point slope form this time (although you could still use slope-intercept form and solve for the $y$-intercept). For point-slope form we need the slope and any point. We know that $m=-2$ and we can use the point $(-2,8)$.

$y-y_1=m(x-x_1)$

$y-8=-2(x-(-2))$

$y-8=-2(x+2)$

$y-8=-2x-4$

$y=-2x+4$

Remember this is $f^{-1}(x)$, so we’ll write

$f^{-1}(x)=-2x+4$

Now we swap $x$ and $y$ to get $f(x)$.

$x=-2y+4$

Solve for $y$.

$x-4=-2y$

$y=-\frac{1}{2}x+2$

Replace $y$ with $f(x)$.

$f(x)=-\frac{1}{2}x+2$

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As you can see, there’s more than one way to solve these types of problems. Use the way that works best for you.

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