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.
If you like you can also use the points to find the function first and then find its inverse.
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???
As you can see, there’s more than one way to solve these types of problems. Use the way that works best for you.
