Precalculus by Richard Wright

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Pride goes before destruction, a haughty spirit before a fall. Proverbs‬ ‭16‬:‭18‬ ‭NIV‬‬

1-09 Inverse Functions

Imagine a function as a vending machine. You insert some money and out comes a bottle of juice. The drink that comes out of the machine is a function of the amount of money that is inserted into the machine. Now imagine a vending machine that works backwards. Something is put into the machine, and it gives out money. In 2009, EcoATM starting putting such machines in stores. A used cellphone is put into the backward vending machine, and it gives you cash.

The vending machine and backward vending machine are examples of an inverse relationship. An inverse function interchanges the input and the output. A regular vending machine inputs money and outputs a product such as a phone. An inverse vending machine inputs a product such as a phone and outputs money.

Verify That Two Functions Are Inverse Functions

Imagine you are going to travel to Canada for a weekend of hiking, but you would like nice weather. You check the weather forecast, but it is given in degrees Celsius. Being an American, you are used to degree Fahrenheit. But, you do remember the conversion from Fahrenheit to Celsius.

$$ C = \frac{5}{9}(F - 32) $$

You start looking at the first day. To find the temperature in Fahrenheit, you decide to plug in Friday's high of 12°C and solve for F.

$$ 12 = \frac{5}{9}(F - 32) $$

$$ \frac{108}{5} = F - 32 $$

$$ F = \frac{108}{5} + 32 ≈ 53.6°F $$

Slightly cool, but a decent temperature for a nice hike. You start to work on Friday night's temperature, but then notice that you would have to do the same algebra steps for each of the six temperatures. It would be easier to solve the formula for F first, then simply plug in the numbers.

$$ C = \frac{5}{9}(F - 32) $$

$$ \frac{9}{5}C = F - 32 $$

$$ F = \frac{9}{5}C + 32 $$

Now you can just plug in the temperatures in Celsius and get the corresponding temperatures in Fahrenheit. This new formula for converting Celsius to Fahrenheit is the inverse of the Fahrenheit to Celsius formula. Inverse functions switch the inputs and outputs.

If the function is named in function notation, f(x), then the inverse function is named f⁻¹(x) which is read "inverse f of x." Beware the −1 is not an exponent and does not mean \(\frac{1}{f(x)}\) because that is the reciprocal and not the inverse.

Inverses are a type of opposite and undo each other. The inverse cancels out the original leaving only the identity function, x.

(f⁻¹ ∘ f)(x) = f⁻¹(f(x)) = x

Some functions do not have inverse functions. Only functions where each input is paired with a unique output have inverse functions. In these one-to-one functions, each x is paired with only one y, and each y in the range is paired with only one x.

Find Domain and Range of Inverse Functions

Inverse functions switch the input and output, so the domain and range are also switched. The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.

When a function has no inverse function because it is not one-to-one, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of \(f(x) = \sqrt{x}\) is f⁻¹ = x², because a square “undoes” a square root. But the square is only the inverse of the square root on the domain [0, ∞), since that is the range of \(f(x) = \sqrt{x}\).

From the other side of the same problem, starting with the square (parent quadratic) function f(x) = x². There is a problem creating the inverse, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and −3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.

To determine if a function is one-to-one and has an inverse function, use the horizontal line test. It says that a function is one-to-one if no horizontal line touches the graph more than once.

In many cases, if a function is not one-to-one, it can still be restricted to a part of its domain on which it is one-to-one. For example, the restricted version of the square function f(x) = x² with its domain limited to [0, ∞) is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

Find Inverse Functions

Inverses switch the x and y-coordinates. To find an inverse function, start by rewriting the f(x) as y. Then interchange the x's and the y's. Next, solve for y. Finally, rewrite the y as f⁻¹(x).

Evaluate the Inverse of a Function From a Graph of the Original Function

Remember from lesson 1-04 that the domain of a function is the part of the x-axis covered by the graph. Because the inverse switches the x and y, the domain of the inverse of a function is the part of the y-axis covered by the graph. The domain and range of inverses are switched. Similarly, a function can be evaluated for an x-value by reading the corresponding y-value from the graph. To evaluate an inverse from the original function's graph, find the input on the y-axis of the graph, then read the corresponding output from the x-axis.

Example 7: Evaluate a Function and Its Inverse from a Graph at Specific Points

Use the graph of f(x) to find f(3) and f⁻¹(2).

Solution

To evaluate f(3), find 3 on the x-axis and find the corresponding output value on the y-axis. The point (3, 1) tells indicates that f(3) = 1.

To evaluate f⁻¹(2), remember that the x and y's are switched for the inverse. Find 2 on the y-axis and find the corresponding value x-axis. The point (4, 2) on the original graph means the point on the inverse is (2, 4). It indicates that f⁻¹(2) = 4.

Graphs of Inverse Functions

Now it is time to explore the graphs of functions and their inverses further. Use the quadratic function f(x) = x² restricted to the domain [0, ∞) so that it is one-to-one, and graph it as in figure 5.

An inverse function, switches the x and y-coordinates. Take each of the points of the graph in figure 5 and exchange the x and y-coordinates so that (2, 4) becomes (4, 2), (3, 9) becomes (9, 3), and so on. This produces the graph in figure 6. Notice that the graph is reflected over the line y = x.

This relationship is true for all inverses. The graphs of a function and its inverse are reflections over the line y = x.

Example 8: Find the Inverse of a Function Graphically

If f(x) is in figure 7, sketch a graph of f⁻¹(x).

Solution

The inverse graph is a reflection over the line y = x, so find a lot of points and move them the same distance onto the other side of the line.

OR the inverse switches the x and y-coordinates. So, take a bunch of points of f(x) and switch their coordinates as in Table 1. Then graph the new points.

Table 1

Point on f(x)	Point on f−1(x)
(4, 2) →	(2, 4)
(2, 1) →	(1, 2)
(1, 0) →	(0, 1)
(1/2, −1) →	(−1, 1/2)
(1/4, −2) →	(−2, 1/4)

Try It 5

If f(x) is in the graph below, sketch a graph of f⁻¹(x).

Solution

Practice Exercises

1. How do you find the inverse of a function algebraically?

Use function composition to verify that f(x) and g(x) are inverse functions.

2. \(f(x) = \sqrt[5]{x + 1}\) and g(x) = x⁵ − 1
3. f(x) = 2x + 1 and \(g(x) = \frac{x-1}{2}\)
4. f(x) = (x + 1)² and \(g(x) = \sqrt{x} - 1\)
5. f(x) = 3 − x and g(x) = −x + 3
6. \(f(x) = \frac{2x+3}{5x+4}\) and \(g(x) = \frac{3-4x}{5x-2}\)

Find f⁻¹(x) algebraically for each function.

7. f(x) = 2x − 3
8. \(f(x) = 2 − \sqrt[3]{x}\)
9. \(f(x) = \frac{x}{x - 4}\)
10. f(x) = 2x², x ≥ 0
11. f(x) = (x + 3)², x ≤ −3
12. Use the graph of f(x) to graph the inverse.
    

Problem Solving

13. The kinetic energy of a 70 kg person is K(v) = 35v² where v is the speed of the object. Find the inverse of the function and explain its meaning. (Note: Speed is always greater than or equal to zero.)
14. A function F(t) gives the number of flowers, F, a bee has visited in time, t. What would be a good name of the inverse of this function, and what is its meaning?

Mixed Review

15. (1-08) Given f(x) = 2x and g(x) = x² + 3, find f + g, f − g, fg, and \(\frac{f}{g}\).
16. (1-07) Describe how the formula is a transformation of a parent function: m(t) = 4(2 − t)³
17. (1-06) Sketch a graph of the piecewise function: \(p(x) = \left\{\begin{align} |x + 2|, \text{ if } x ≤ 1 \\ -(x - 1)^2 + 3, \text{ if } x > 1\end{align}\right.\)
18. (1-05) Write the domain and range of the function shown in the graph using interval notation.
    
19. (1-04) Evaluate f(2), f(−1), and f(t + 2) for f(x) = −2(x + 1)².
20. (1-03) Write an equation for a line perpendicular to y = 2x + 3 and passing through the point (0, 1).

Answers

1. 1. Make sure f is a one-to-one function. 2. Rewrite the f(x) as y. 3. Interchange the x's and the y's. 4. Solve for y. 5. Rewrite the y as f⁻¹(x).
2. f(g(x)) = x
3. f(g(x)) = x
4. f(g(x)) = x
5. f(g(x)) = x
6. f(g(x)) = x
7. \(f^{−1}(x) = \frac{x + 3}{2}\)
8. f⁻¹(x) = (2 − x)³
9. \(f^{-1}(x) = \frac{4x}{x-1}\)
10. \(f^{-1}(x) = \sqrt{\frac{x}{2}}\)
11. \(f^{-1}(x) = -\sqrt{x} - 3\)
12. 
13. \(v(K) = \sqrt{\frac{K}{35}}\); The inverse gives the speed of a given kinetic energy
14. t(F); the time it takes the bee to visit F flowers
15. x² + 2x + 3; −x² + 2x − 3; 2x³ +6x; \(\frac{2x}{x^2+3}\)
16. Vertical stretch by a factor of 4, reflected over the y-axis, shifted right 2.
17. 
18. Domain: [−3, 3]; Range: [−3, 3]
19. −18, 0, −2t² − 12t − 18
20. \(y = -\frac{1}{2}x + 1\)