A function is like an equation that relates an input to an output. We commonly express functions in the form:
$f(x)$f(x)$=$=...
A linear function is a relationship between two variables that, when graphed, will be in a straight line.
An inverse function is a function that reverses another function. It sounds a bit confusing so let's look at how to find inverses in a bit more detail.
The inverse of a pair of coordinates (a point) can be thought of as the corresponding point if the point was reflected along the line $y=x$y=x.
For example, the line $y=x$y=x has been graphed and I have plotted the point $\left(-2,3\right)$(−2,3) in blue.
So to find the inverse, I reflect across this line. To do this I reverse the coordinates. This mean that the $y$y becomes the $x$x and the $x$x becomes the $y$y. So $\left(-2,3\right)$(−2,3) becomes $\left(3,-2\right)$(3,−2).
This is shown on the graph in green.
The reflection is evident by the fact that I have crossed the line $y=x$y=x at $90^\circ$90°.
We'll run through the process of finding an inverse function it using an example: $f(x)$f(x)$=$=$3x+6$3x+6
1. Start with the original function, substituting $y$y for $f(x)$f(x):
$y=3x+6$y=3x+6
2. Rearrange the equation to make $x$x the subject:
$y$y | $=$= | $3x+6$3x+6 |
$y-6$y−6 | $=$= | $3x$3x |
$\frac{y-6}{3}$y−63 | $=$= | $x$x |
$x$x | $=$= | $\frac{y-6}{3}$y−63 |
3. Switch the places of $x$x and $y$y in the equation you found in step 2:
We started with $x=\frac{y-6}{3}$x=y−63, so when we switch the variables to make it $y=\frac{x-6}{3}$y=x−63
So the inverse function of $f(x)$f(x)$=$=$3x+6$3x+6 is $f(x)$f(x)$=$=$\frac{x-6}{3}$x−63
Now let's look at some examples.
Define $y$y, the inverse of $f\left(x\right)=-\frac{1}{8}x+10$f(x)=−18x+10.
Find the inverse of $3x+48=8y$3x+48=8y.
If $f\left(x\right)$f(x)$=$=$kx$kx $-$− $7$7 and $f^{-1}\left(x\right)$f−1(x)$=$=$2x$2x$+$+$14$14, solve for the value of $k$k.