Inverse operations are operations that 'undo' each other - for example, addition and subtraction, or multiplication and division. We can extend this concept to find the inverse of an entire function or relation.

Complete the table below showing the relationship between the length of the sides of a cube and the volume of the cube.

Length of sides | Volume of the cube |
---|---|

1 | |

2 | |

3 |

Volume of the cube | Length of sides |
---|---|

1 | |

8 | |

27 |

Then, graph the data in the table on the same set of axes using a different color to draw each graph.

Inverses are useful for determining the input of a relation if the outputs are known. Consider a situation where a plane is traveling at a constant speed, and we want to know how long the plane has been flying over certain distances. Rather than using the function d\left(t\right)=rt and dividing by the rate to find the time for each of the distances, we can simply rewrite the equation as t=\dfrac{d}{r}. This is the inverse relation of d\left(t\right).

Geometrically, this means that the relation and its inverse are mirror images of each other across the line y=x.

x | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
---|---|---|---|---|---|---|---|

f\left(x\right) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

g\left(x\right) | -6 | -4 | -2 | 0 | 2 | 4 | 6 |

The function inverse to f\left(x\right) is denoted f^{-1}\left(x\right), so if \left(a,b\right) is an element of f, then \left(b,a\right) is an element of f^{-1}.

Complete the tables. State which relations are inverses.

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

f\left(x\right)=2x^3 |

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

g\left(x\right)=\frac{1}{2}x^3 |

x | -16 | -2 | 0 | 2 | 16 |
---|---|---|---|---|---|

h\left(x\right)=\sqrt[3]{\frac{x}{2}} |

x | -1 | -\frac{1}{8} | 0 | \frac{1}{8} | 1 |
---|---|---|---|---|---|

j\left(x\right)=2\sqrt[3]{x} |

Worked Solution

Find the inverse of each function using the same representation.

a

x | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|

f\left(x\right)=(x+1)^2 +5 | 5 | 6 | 9 | 14 | 21 |

Worked Solution

b

f(x)=\sqrt[3]{x}

Worked Solution

c

f(x)=\sqrt{x}+3

Worked Solution

Idea summary

We can verify a relation's inverse by graphing the relations to show the two relations are reflected across the line y=x.

We can find the inverse by:

- Swapping x and y in a table of values
- Graphically swapping the x and y coordinates

Explore the applet by choosing a function and dragging the slider to produce the function's inverse.

- The inverse of a linear function is also always a function. What do you notice about the inverse of each of these parent functions?
- How does restricting the domain make a relation become a function?

Any function can be reflected across the line y=x, but not all reflections will satisfy the definition of a function.

A function such as f(x)=x^2 does not have an inverse function. If we reflect f(x)=x^2 across the line y=x we will get a relation that is not a function.

We can restrict the domain of the function in order for the inverse function to exist and pass the vertical line test.

Restrict to x\geq0, the inverse is {f^{-1}(x)=\sqrt{x}}

Restrict to x\leq0, the inverse is {f^{-1}(x)=-\sqrt{x}}

When we reflect a function over the line y=x, we are effectively switching the x and y-values. To find the inverse algebraically, we swap x and y in the equation, then solve for y to get f^{-1}(x).

Write f\left(x\right) as y

Swap x and y

Solve for y

Replace y with f^{-1}(x)

Swapping the x and y variables in a relationship will exchange the coordinates for any point on the graph. Thus, the domain and range will be swapped in an inverse relation compared to the original relation. That is, the domain of the function is the same as the range of the inverse function and the range of the function is the same as the domain of the inverse function.

Consider the graphs of f\left(x\right), g\left(x\right) and h\left(x\right) and determine if they are invertible functions. Explain how you know.

a

f\left(x\right)

Worked Solution

b

g\left(x\right)

Worked Solution

c

h\left(x\right)

Worked Solution

For each of the following functions:

- Determine an expression for the inverse relation.
- State whether or not the inverse is a function.
- If the inverse is not a function, find a restricted domain for the function under which the inverse is a function.

a

y=7x- 4

Worked Solution

b

y = \left(x-3\right)^2-5

Worked Solution

c

f \left(x \right) = \sqrt{x-3} +2 for x\geq3

Worked Solution

Idea summary

- In order to be inverses, the domain and range of a function must be the same as the range and domain of the inverse function.
- We can find the inverse algebraically by swapping x and y and solving the equation for y

To verify that two functions, f(x) and g(x), are inverses of each other, we can use function composition. Given f(x) and g(x), if:

- f(g(x))=x
- g(f(x))=x

Then, we can say that f(x) and g(x) are inverse functions.

We must check the composition both ways and if both equal x, then the two functions are inverses. If only one of the compositions is equal to x, the functions are not necessarily inverses.

Determine if the pair of given functions are inverses. Justify your answer.

a

f(x)=4x-2 and g(x)=\dfrac{x+2}{4}

Worked Solution

b

f(x)=5(x+4)^3 and g(x)=\dfrac{1}{5}\sqrt[3]{x}-4

Worked Solution

c

f(x)=\sqrt{x-5} and g(x)=x^2 +5, x \geq 5

Worked Solution

Idea summary

Two functions, f(x)and g(x) are inverse functions, if both f(g(x))=x and g(f(x))=x.