Learning objectives
Explore the applet by choosing a function and dragging the slider to produce the function's inverse.
Any function can be reflected across the line y=x to create an inverse relation, but not all inverse relations will satisfy the definition of a function.
All inverse functions are inverse relations, much like all functions are relations, but not all inverse relations are inverse functions.
Functions whose inverse is also a function are called one-to-one functions and have special properties.
Note that in order to be a function, a relation maps each input to exactly one output. A one-to-one function includes the additional criteria that the function must map each output to exactly one input.
For each of the following functions:
y = \left(x-3\right)^2-5
y=\left(x- 4 \right) \left(x^2 +4x + 16 \right)
f \left(x \right) = \dfrac{4}{x+5}
Consider the function shown in the table below.
x | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
f\left(x\right) | 32 | 16 | 8 | 4 | 2 | 1 |
Determine if it is possible for an inverse function to exist without restricting the domain.
Determine the value of f^{-1}\left(8\right).
Determine the value of x that makes f^{-1}\left(x\right)=1.
Find the inverse function of y=-8x+6.
We can determine if a function's inverse is also a function by using the horizontal line test on the function. If the inverse is not a function, we can restrict the domain of the given function.
A function, f(x), is a relation where for any value of the independent variable there is only one value of the dependent variable.
An inverse function, f^{-1}(x), is an inverse relation which is also a function.
We can find inverse functions and relations by swapping the position of the dependent and independent variables in the equation and then rearranging the equation to make the new dependent variable the subject.
Graphs represent relations in the same way that equations do. Each point on the graph has an x-value and a y-value which satisfies the rules of the relation.
If a graph represents a function then each x-value has only one corresponding y-value. This means that any vertical line will intersect the graph at at most one point. This gives us the vertical line test. If we can draw a vertical line which intersects the graph more than once then the graph does not represent a function.
When a function has an inverse each value of the dependent variable has only one possible value for the independent variable. On a graph that means that each y-value has only one corresponding x-value and so any horizontal line will intersect the graph at at most one point. This gives us the horizontal line test. If we can draw a horizontal line which intersects the graph more than once then the function does not have an inverse function.
When we want to find the equation of an inverse function, we swap the positions of the independent and dependent variables. Equivalently, to find the graph of an inverse function, we swap the x-coordinates with y-coordinates. This is the same as reflecting the graph about the line y=x.
Consider the graphs of f\left(x\right), g\left(x\right) and h\left(x\right) and determine if they have an inverse function without any domain restrictions. Explain how you know.
f\left(x\right)
g\left(x\right)
h\left(x\right)
Do the following graphs have inverse functions?
The lines y=5x (labelled B) and y=x (labelled A) have been plotted below.
By reflecting y=5x about the line y=x, plot the graph of the inverse of y=5x.
The vertical line test says that any vertical line will intersect a function at at most one point.
The horizontal line test says that if a function has an inverse function any horizontal line will intersect the function at at most one point.
The graph of an inverse function will be the graph of the original function reflected about the line y=x.
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.
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).
We can find the algebraic equation for the inverse relation by completing the following steps:
Write f\left(x\right) as y
Swap x and y
Solve for y
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 switched in an inverse relation compared to the original relation.
Geometrically, this means that the relation and its inverse are mirror images of each other across the line y=x.
The volume of a sphere is modeled by the equation V\left(r\right)=\dfrac{4}{3}\pi r^3.
For each of the following volumes, find the radius of the corresponding sphere.
Describe a more efficient way to find the radius when the volume is known.
Explain how your answer to part (b) relates to the equation for volume.
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} |
Consider the absolute value function y=\left|x+3\right|+6.
State the domain and range of the inverse relation.
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: