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Questions and Answers
If a function 𝑓 is invertible and (𝑎, 𝑏) is on its graph, what is the corresponding point on the graph of 𝑓 −1?
What does the point (𝑏, 𝑎) represent on the graph of an inverse function if (𝑎, 𝑏) is a point on the graph of function 𝑓?
The relationship between the graphs of a function 𝑓 and its inverse 𝑓 −1 can be described as:
If a function 𝑓 has an inverse, what can be inferred about the function's graph?
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Given a point (𝑎, 𝑏) on the graph of an invertible function, what is the best method to find its equivalent point on the graph of 𝑓 −1?
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What is the first step in finding the inverse of a function?
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When given the function g(x) = 2/(7x + 5), what do you obtain after interchanging variables in the first step?
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What is the result of solving 7y + 5 = 2x for y in terms of x?
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In the function h(x) = 4 - 3x, what is the correct expression for h^{-1}(x) after finding its inverse?
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After completing the steps to find the inverse of f(x) = 4050x, what is the expression for f^{-1}(x)?
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What is achieved in the final step of finding the inverse function?
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Which equation represents the proper interchange of x and y for the function given as y = 4 - 3x?
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In the function g(x) = 2/(7x + 5), what is the form of g^{-1}(x) after solving?
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How do you interpret f^{-1}(y) = x for a given function f?
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Which step is involved when solving the equation 3y = 4 - x to isolate y?
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Study Notes
Finding the Inverse of a Function
- To find the inverse of a function, follow these steps:
- Let y equal to the function f(x).
- Interchange the variables x and y.
- Solve for y in terms of x.
- Replace y by f-1(x).
- Note that f-1(y) = x, meaning the inverse function undoes what the original function does to a number within its domain.
Examples of Finding the Inverse
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Example 1: Find the inverse of f(x) = 4,050x.
- Step 1: Let y = 4,050x
- Step 2: Interchange x and y, so x = 4,050y
- Step 3: Solve for y, resulting in y = x/4,050
- Step 4: Replace y with f-1(x), giving f-1(x) = x/4,050
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Example 2: Find the inverse of g(x) = (7x+5)/2.
- Step 1: Let y = (7x + 5)/2
- Step 2: Interchange x and y, resulting in x = (7y + 5)/2
- Step 3: Solve for y, giving y = (2x - 5)/7
- Step 4: Replace y with g-1(x), resulting in g-1(x) = (2x - 5)/7
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Example 3: Find the inverse of h(x) = 4 - 3x
- Step 1: Let y = 4 - 3x
- Step 2: Interchange x and y, giving x = 4 - 3y
- Step 3: Solve for y, resulting in y = (4 - x)/3
- Step 4: Replace y with h-1(x), giving h-1(x) = (4 - x)/3
Graphing Inverse Functions
- If a function f is invertible and (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f-1. This means the graph of f-1 is the reflection of the graph of f across the line y = x.
- In other words, if you find a point (a, b) on f, you will find a point (b, a) on its inverse f-1.
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Description
This quiz explores the process of finding the inverse of a function. You will learn how to interchange variables and solve for y in terms of x through a variety of examples. Test your understanding of function inverses and their properties.
