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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?
If a function 𝑓 is invertible and (𝑎, 𝑏) is on its graph, what is the corresponding point on the graph of 𝑓 −1?
- (−𝑏, 𝑎)
- (𝑎, 𝑏)
- (𝑏, 𝑎) (correct)
- (𝑏, −𝑎)
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?
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?
- A point on the graph of 𝑓 −1 (correct)
- A point on the graph of 𝑓
- A non-existent point
- A point outside the domain of 𝑓 −1
The relationship between the graphs of a function 𝑓 and its inverse 𝑓 −1 can be described as:
The relationship between the graphs of a function 𝑓 and its inverse 𝑓 −1 can be described as:
- They are symmetrical about the x-axis.
- They are identical graphs.
- They are reflections over the y-axis.
- They are symmetrical about the line 𝑦 = 𝑥. (correct)
If a function 𝑓 has an inverse, what can be inferred about the function's graph?
If a function 𝑓 has an inverse, what can be inferred about the function's graph?
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?
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?
What is the first step in finding the inverse of a function?
What is the first step in finding the inverse of a function?
When given the function g(x) = 2/(7x + 5), what do you obtain after interchanging variables in the first step?
When given the function g(x) = 2/(7x + 5), what do you obtain after interchanging variables in the first step?
What is the result of solving 7y + 5 = 2x for y in terms of x?
What is the result of solving 7y + 5 = 2x for y in terms of x?
In the function h(x) = 4 - 3x, what is the correct expression for h^{-1}(x) after finding its inverse?
In the function h(x) = 4 - 3x, what is the correct expression for h^{-1}(x) after finding its inverse?
After completing the steps to find the inverse of f(x) = 4050x, what is the expression for f^{-1}(x)?
After completing the steps to find the inverse of f(x) = 4050x, what is the expression for f^{-1}(x)?
What is achieved in the final step of finding the inverse function?
What is achieved in the final step of finding the inverse function?
Which equation represents the proper interchange of x and y for the function given as y = 4 - 3x?
Which equation represents the proper interchange of x and y for the function given as y = 4 - 3x?
In the function g(x) = 2/(7x + 5), what is the form of g^{-1}(x) after solving?
In the function g(x) = 2/(7x + 5), what is the form of g^{-1}(x) after solving?
How do you interpret f^{-1}(y) = x for a given function f?
How do you interpret f^{-1}(y) = x for a given function f?
Which step is involved when solving the equation 3y = 4 - x to isolate y?
Which step is involved when solving the equation 3y = 4 - x to isolate y?
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
- 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
- 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
- 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.
