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The idea of inverse functions is that if you input a value into one function, then take that result and input it into the inverse function, you should get the original value back. You could also think of the inverse function undoing the original function. Figure 1 should help you visualize what is going on.

Figure 1

The idea is that \(x\) is input to the function \(f(x)\) with output y. This \(y\) is then the input of the inverse function, whose output is the orginal \(x\) that we started with. Pretty cool, eh? A couple of comments are in order.
1. We use the notation \(f^{-1}(y)\) to describe the inverse of \(f(x)\). In this context, \(f^{-1}(y) \neq 1/f(y)\). You can usually tell by the context what is meant when you run across this notation.
2. This loop works in both directions, i.e. if you reverse the direction of the arrows and input x into the inverse function first, the result is the same.

Here is a great video explaining inverse functions and where we are going with them.

You are probably familiar with the vertical line test that tells you whether the graph is a function or not. Well, a similiar rule holds called the horizontal line test. When you have a graph and you draw horizontal lines, if all horizontal lines pass through the graph only once, then the graph is invertible, i.e. an inverse exists. If you can draw at least one line that passes through the graph more than once, then the graph is NOT invertible and no inverse exists.

One thing to note is that this tells you whether or an inverse exists but not how to find it. In fact, you may not be able to find the inverse even if the horizontal line test tells you that one exists.

Here is a good video with lots of examples, explaining the horizontal line test. He also mentions the vertical line test, which helps you see the parallel.

When a graph does not pass the horizontal line test, we can often restrict the domain, so that the graph does pass the horizontal line test and then have an inverse. (This is especially helpful in the case of trigonometric functions.) For example, you know that the parabola \(y=x^2\) does not pass the horizontal line test. However, we can restrict the domain by requiring that \(x > 0\). We now have a graph that passes the horizontal line test and therefore has an inverse.

Finding an inverse of a function is pretty straightforward. There are some specific steps to follow but the idea is that you change all x's to y's and all y's to x's and then solve the new equation for y. Rather than going into a lot of detail, this video explains it very well.

This video goes into a lot of detail with inverse functions and gives the entire picture. He also has some really good examples.

Something you always want to do in math is try to check your answer. The way you check your answer after calculating the inverse is to form two composite functions and see what you get. For example, if your original function was \(f(x)\), we can write the inverse as \(f^{-1}(x)\). The notation here can get confusing. Since the context is inverse functions, \(f^{-1}(x) \neq 1/f(x)\).

The two composite functions are \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). In both cases, the result should be x, if they are truly inverses of each other. Note that you need to calculate BOTH composite functions, since, if you made a mistake, it is possible for one to evaluate to x and the other not be equal to x.

If you need to review composite functions, you can find more explanation and practice problems on the composite functions page.

Figure 2

The graph of an inverse function is interesting when compared to the orginal function. The graphs are reflected about the line \(y=x\). If you think about it, it makes sense. After all, in the discussion above, we switch the x's and y's as one of the steps when finding an inverse. Figure 2 shows an example. The two curved graphs are inverses of each other. The black straight line is the line \(y=x\). Notice the nice symmetry across the line.

Okay, time for some practice problems.

Practice

Unless otherwise instructed, find the inverse of these functions without using a calculator and give your answers in exact, simplified form.

• Practice 1559
• Solution

\(f(x)=\sqrt{x+4}-3\)

Problem Statement

Find the inverse of the function \(f(x)=\sqrt{x+4}-3\) without using a calculator and give your answer in exact, simplified form.

Solution

PatrickJMT - 1559 video solution

video by PatrickJMT

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• Practice 1560
• Solution

\(\displaystyle{y=\frac{5x-3}{2x+1}}\)

Problem Statement

Find the inverse of the function \(\displaystyle{y=\frac{5x-3}{2x+1}}\) without using a calculator and give your answer in exact, simplified form.

Solution

PatrickJMT - 1560 video solution

video by PatrickJMT

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• Practice 1566
• Solution

\(f(x)=\sqrt{x-2}\)

Problem Statement

Find the inverse of the function \(f(x)=\sqrt{x-2}\) without using a calculator and give your answer in exact, simplified form.

Solution

Krista King Math - 1566 video solution

video by Krista King Math

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• Practice 1567
• Answer
• Solution

\( f(x)=\sqrt{-1-x} \)

Problem Statement

Find the inverse of the function \( f(x)=\sqrt{-1-x} \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1} = -x^2-1 \) for \( x \geq 0 \)

Problem Statement

Find the inverse of the function \( f(x)=\sqrt{-1-x} \) without using a calculator and give your answer in exact, simplified form.

Solution

Rewrite the function using \(y\) instead of \(f(x)\).

\( y=\sqrt{-1-x} \)

Change \(x \to y \) and \( y \to x \) and solve for \(y\).

\( x = \sqrt{-1-y} \)

\( x^2 = -1-y \)

Squaring both sides is okay as long as we remember that \(x\) is non-negative.

\( y = -x^2-1 \)

We will name this function \(g(x)\).

\( g(x) = -x^2-1 \)

At this point, we have \(g(x)\), which is the inverse function of \(f(x)\) but let's verify our answer by finding BOTH composite functions.

\( f \circ g = \sqrt{-1-[-x^2-1]} \)

\( \sqrt{ -1+x^2+1 } = \sqrt{x^2} = x \)

Okay, so \( f \circ g \) is good. Now find \( g \circ f \)

\( g \circ f = -(\sqrt{-1-x})^2-1 \)

\( -(-1-x)-1 = 1+x-1 = x \)

So \( g \circ f \) is also good.

So our inverse function is \( f^{-1} = g(x) = -x^2-1 \) for \( x \geq 0\)

We must specify that \( x \geq 0\) in our answer.

Even though we were not asked to do so, I was interested in what the plots looked like, so I plotted the graphs of \(f(x)\) and \(g(x)\) as well as \(y=x\). The plot is shown below.

Final Answer

\( f^{-1} = -x^2-1 \) for \( x \geq 0 \)

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• Practice 2903
• Answer
• Solution

\( f(x) = 2x - 7 \)

Problem Statement

Find the inverse of the function \( f(x) = 2x - 7 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = (x + 7)/2 \)

Problem Statement

Find the inverse of the function \( f(x) = 2x - 7 \) without using a calculator and give your answer in exact, simplified form.

Solution

2903 video solution

Final Answer

\( f^{-1}(x) = (x + 7)/2 \)

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• Practice 2904
• Answer
• Solution

\( f(x) = x^3 + 8 \)

Problem Statement

Find the inverse of the function \( f(x) = x^3 + 8 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = \sqrt[3]{x-8} \)

Problem Statement

Find the inverse of the function \( f(x) = x^3 + 8 \) without using a calculator and give your answer in exact, simplified form.

Solution

2904 video solution

Final Answer

\( f^{-1}(x) = \sqrt[3]{x-8} \)

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• Practice 2905
• Answer
• Solution

\( f(x) = \sqrt{x+2} - 5 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt{x+2} - 5 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = x^2 + 10x + 23 \) The instructor does not say anything about the domain of the inverse function. He needs to specify that for \( f^{-1}(x) \) the domain is \( x \geq -5 \). This is important since your instructor may take off points if you do not specify the domain here.
How do you get this domain? Well, we got from the graph but if you don't have the graph, you know the domain of \(f(x)\) is \(x \geq -2 \) since you cannot take the square root of negative numbers. Now, the range of \(f(x) \) is \( y \geq -5 \). In this case, the range of \(f(x) \) becomes the domain of \(f^{-1}(x)\). So the domain of the invers function is \( x \geq -5 \).

Problem Statement

Find the inverse of the function \( f(x) = \sqrt{x+2} - 5 \) without using a calculator and give your answer in exact, simplified form.

Solution

2905 video solution

Final Answer

\( f^{-1}(x) = x^2 + 10x + 23 \) The instructor does not say anything about the domain of the inverse function. He needs to specify that for \( f^{-1}(x) \) the domain is \( x \geq -5 \). This is important since your instructor may take off points if you do not specify the domain here.
How do you get this domain? Well, we got from the graph but if you don't have the graph, you know the domain of \(f(x)\) is \(x \geq -2 \) since you cannot take the square root of negative numbers. Now, the range of \(f(x) \) is \( y \geq -5 \). In this case, the range of \(f(x) \) becomes the domain of \(f^{-1}(x)\). So the domain of the invers function is \( x \geq -5 \).

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• Practice 2906
• Answer
• Solution

\( f(x) = \sqrt[3]{x+4} - 2 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt[3]{x+4} - 2 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = (x+2)^3 - 4 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt[3]{x+4} - 2 \) without using a calculator and give your answer in exact, simplified form.

Solution

2906 video solution

Final Answer

\( f^{-1}(x) = (x+2)^3 - 4 \)

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• Practice 2907
• Answer
• Solution

\(\displaystyle{ f(x) = \frac{3x-7}{4x+3} }\)

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{3x-7}{4x+3} }\) without using a calculator and give your answer in exact, simplified form.

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{3x+7}{3-4x} }\) The instructor does not specify the domain of this inverse function. We need to restrict the domain to all real numbers except \( x = 3/4 \) and \( x = -3/4 \). The first number occurs when the denominator of the inverse function is zero. The second number occurs when the denominator of \( f(x) \) is zero. Both values need to specified. However, check with your instructor to see what they require.

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{3x-7}{4x+3} }\) without using a calculator and give your answer in exact, simplified form.

Solution

2907 video solution

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{3x+7}{3-4x} }\) The instructor does not specify the domain of this inverse function. We need to restrict the domain to all real numbers except \( x = 3/4 \) and \( x = -3/4 \). The first number occurs when the denominator of the inverse function is zero. The second number occurs when the denominator of \( f(x) \) is zero. Both values need to specified. However, check with your instructor to see what they require.

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• Practice 2908
• Answer
• Solution

\(\displaystyle{ f(x) = \frac{x+2}{x} }\)

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{x+2}{x} }\) without using a calculator and give your answer in exact, simplified form.

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{2}{x-1} }\) with domain \( \{ x | x \in \mathbb{R}, x \neq 0, x \neq 1 \} \)

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{x+2}{x} }\) without using a calculator and give your answer in exact, simplified form.

Solution

2908 video solution

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{2}{x-1} }\) with domain \( \{ x | x \in \mathbb{R}, x \neq 0, x \neq 1 \} \)

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• Practice 2909
• Answer
• Solution

\(\displaystyle{ f(x) = \frac{2x+5}{3x-1} }\)

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{2x+5}{3x-1} }\) without using a calculator and give your answer in exact, simplified form.

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{3x+5}{3x-2} }\) with domain \( \{ x | x \in \mathbb{R}, x \neq 1/3, x \neq 2/3 \} \)

Problem Statement

Find the inverse of the function \(\displaystyle{ f(x) = \frac{2x+5}{3x-1} }\) without using a calculator and give your answer in exact, simplified form.

Solution

2909 video solution

Final Answer

\(\displaystyle{ f^{-1}(x) = \frac{3x+5}{3x-2} }\) with domain \( \{ x | x \in \mathbb{R}, x \neq 1/3, x \neq 2/3 \} \)

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• Practice 2910
• Answer
• Solution

\( f(x) = \sqrt{2x-6} \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt{2x-6} \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = x^2 / 2 + 3 \) for \( x > 0 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt{2x-6} \) without using a calculator and give your answer in exact, simplified form.

Solution

2910 video solution

Final Answer

\( f^{-1}(x) = x^2 / 2 + 3 \) for \( x > 0 \)

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• Practice 2911
• Answer
• Solution

\( f(x) = \sqrt[3]{x-4} + 1 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt[3]{x-4} + 1 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = (x-1)^3 + 4 \)

Problem Statement

Find the inverse of the function \( f(x) = \sqrt[3]{x-4} + 1 \) without using a calculator and give your answer in exact, simplified form.

Solution

2911 video solution

Final Answer

\( f^{-1}(x) = (x-1)^3 + 4 \)

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• Practice 2912
• Answer
• Solution

\( f(x) = e^{3x+1} - 5 \)

Problem Statement

Find the inverse of the function \( f(x) = e^{3x+1} - 5 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = (1/3)\ln(x-5) - 1/3 \) with domain \( x > 5 \)

Problem Statement

Find the inverse of the function \( f(x) = e^{3x+1} - 5 \) without using a calculator and give your answer in exact, simplified form.

Solution

2912 video solution

Final Answer

\( f^{-1}(x) = (1/3)\ln(x-5) - 1/3 \) with domain \( x > 5 \)

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• Practice 2913
• Answer
• Solution

\( f(x) = 2x-3 \)

Problem Statement

Find the inverse of the function \( f(x) = 2x-3 \) without using a calculator and give your answer in exact, simplified form.

Final Answer

\( f^{-1}(x) = (x+3)/2 \)

Problem Statement

Find the inverse of the function \( f(x) = 2x-3 \) without using a calculator and give your answer in exact, simplified form.

Solution

2913 video solution

Final Answer

\( f^{-1}(x) = (x+3)/2 \)

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