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.
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???
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.