Pure Mathematics 1

Mathematics9709

Pure Mathematics 1

Mathematics9709

The graph of a function and its inverse

You can sketch the graph of an inverse without first finding its formula. The inverse swaps every input and output.

The graph of $\mathrm{f}^{-1}$ is the graph of $\mathrm{f}$ reflected in the line $y=x$ . Every point $(a,b)$ becomes $(b,a)$ , so domain and range swap.

Reflect every point in y = x

The blue curve is f; the green curve is its inverse. The dashed line is the mirror line $y=x$ . The labelled point pair shows the coordinate swap.

f in blue and its inverse in green: each point (a, b) on f goes to (b, a) on the inverse, a mirror image in the dashed line y = x.

Sketch the inverse accurately

Mark the mirror line, swap important points and reflect the curve's shape. A self-inverse graph is unchanged by this reflection.

Worked example

Sketch $\mathrm{f}(x)=(x-1)^2-2$ for $1\le x\le 4$ , and its inverse

On $x\ge 1$ this is the right half of a parabola. It rises from $(1,-2)$ to $(4,7)$ , and is one-one, so an inverse exists.
To draw $\mathrm{f}^{-1}$ , reflect that curve in $y=x$ : the endpoints swap coordinates, $(1,-2)\to(-2,1)$ and $(4,7)\to(7,4)$ .
Plot the mirror image through those two swapped points, and draw the line $y=x$ so the symmetry is visible.

Answer

Reflect the graph of f in

y=x

Sketch checklist

Common mistake

Use

y=x

, not either axis. Show the line and at least one swapped endpoint or key point so the reflection is unambiguous.

Now you try

Your turnreflect in y = x

A function $\mathrm{f}$ is one-one, and the point $(-7,-4)$ lies on its graph. Write down a point that must lie on the graph of $\mathrm{f}^{-1}$ , and state the mirror line that maps one graph onto the other. (9709-style)

Show worked answer

Reflecting in $y=x$ swaps the coordinates, so $(-7,-4)$ on $\mathrm{f}$ becomes $(-4,-7)$ on $\mathrm{f}^{-1}$ .

Answer

Point on the inverse:

(-4,-7)

Mirror line:

y=x

PreviousInverse Functions NextThe Four Function Transformations

The graph of a function and its inverse

You can sketch the graph of an inverse without first finding its formula. The inverse swaps every input and output.

The graph of $\mathrm{f}^{-1}$ is the graph of $\mathrm{f}$ reflected in the line $y=x$ . Every point $(a,b)$ becomes $(b,a)$ , so domain and range swap.

Reflect every point in y = x

The blue curve is f; the green curve is its inverse. The dashed line is the mirror line $y=x$ . The labelled point pair shows the coordinate swap.

f in blue and its inverse in green: each point (a, b) on f goes to (b, a) on the inverse, a mirror image in the dashed line y = x.

Sketch the inverse accurately

Mark the mirror line, swap important points and reflect the curve's shape. A self-inverse graph is unchanged by this reflection.

Worked example

Sketch $\mathrm{f}(x)=(x-1)^2-2$ for $1\le x\le 4$ , and its inverse

On $x\ge 1$ this is the right half of a parabola. It rises from $(1,-2)$ to $(4,7)$ , and is one-one, so an inverse exists.
To draw $\mathrm{f}^{-1}$ , reflect that curve in $y=x$ : the endpoints swap coordinates, $(1,-2)\to(-2,1)$ and $(4,7)\to(7,4)$ .
Plot the mirror image through those two swapped points, and draw the line $y=x$ so the symmetry is visible.

Answer

Reflect the graph of f in

y=x

Sketch checklist

Common mistake

Use

y=x

, not either axis. Show the line and at least one swapped endpoint or key point so the reflection is unambiguous.

Now you try

Your turnreflect in y = x

Show worked answer

Reflecting in $y=x$ swaps the coordinates, so $(-7,-4)$ on $\mathrm{f}$ becomes $(-4,-7)$ on $\mathrm{f}^{-1}$ .

Answer

Point on the inverse:

(-4,-7)

Mirror line:

y=x

PreviousInverse Functions NextThe Four Function Transformations