NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Linear Functions and Inverses

Lesson

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$3`x`+6

1. Start with the original function, substituting $y$`y` for $f(x)$`f`(`x`):

$y=3x+6$`y`=3`x`+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 pronumerals to make it $y=\frac{x-6}{3}$`y`=`x`−63

So the inverse function of $f(x)$`f`(`x`)$=$=$3x+6$3`x`+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`)=−18`x`+10.

Find the inverse of $3x+48=8y$3`x`+48=8`y`.

If $f\left(x\right)$`f`(`x`)$=$=$kx$`k``x` $-$− $7$7 and $f^{-1}\left(x\right)$`f`−1(`x`)$=$=$2x$2`x`$+$+$14$14, solve for the value of $k$`k`.

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions