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Questions and Answers
Which equation represents the correct inverse function for f(x) = 8x - 5?
Which equation represents the correct inverse function for f(x) = 8x - 5?
- f⁻¹(x) = 8(x + 5)
- f⁻¹(x) = 8x + 5
- f⁻¹(x) = (x - 5) / 8
- f⁻¹(x) = (x + 5) / 8 (correct)
What is the notation used to denote the inverse function of f(x)?
What is the notation used to denote the inverse function of f(x)?
- f.inv(x)
- inverse(f(x))
- f*(x)
- f^(-1)(x) (correct)
To find the inverse function of f(x) = x² + 4, what is the first step?
To find the inverse function of f(x) = x² + 4, what is the first step?
- Replace f(x) with y (correct)
- Subtract 4 from both sides
- Replace x with y
- Square both sides
If you need to evaluate f⁻¹(2) for the function f(x) = x / 5 + 1, what is the correct approach?
If you need to evaluate f⁻¹(2) for the function f(x) = x / 5 + 1, what is the correct approach?
When finding the inverse of the composite function g(f(x)), which of the following steps is performed first?
When finding the inverse of the composite function g(f(x)), which of the following steps is performed first?
Flashcards
Inverse Function
Inverse Function
A function that reverses the process of the original function. It takes the output of the original function and returns the input.
Relationship between a Function and its Inverse
Relationship between a Function and its Inverse
A function that takes an input and produces an output, then the inverse function takes the output from the original function and returns the original input.
Finding an Inverse Function
Finding an Inverse Function
To find the inverse of a function, replace f(x) with y, swap all x's and y's, rearrange the equation to solve for y, and then replace y with f⁻¹(x).
Inverse of a Composite Function
Inverse of a Composite Function
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Evaluating an Inverse Function
Evaluating an Inverse Function
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Study Notes
Inverse Functions
- An inverse function reverses the process of the original function.
- If the original function takes an input of 4 and produces an output of 30, the inverse function would take an input of 30 and produce an output of 4.
- The inverse function of f(x) is denoted as f⁻¹(x).
- The function f(x) = 7x + 2 has an inverse function of f⁻¹(x) = (x - 2) / 7.
Finding the Inverse Function
- To find the inverse function, follow these steps:
- Replace f(x) with y.
- Swap all x's with y's and y's with x's.
- Rearrange the equation to make y the subject.
- Replace y with f⁻¹(x).
Examples of Finding Inverse Functions
- Example 1:
- f(x) = 8x - 5
- f⁻¹(x) = (x + 5) / 8
- Example 2:
- f(x) = x / 5 + 1
- f⁻¹(x) = 5(x - 1)
- Example 3:
- f(x) = x² + 4
- f⁻¹(x) = √(x - 4)
Finding the Value of an Inverse Function at a Specific Value
- To evaluate f⁻¹(2), find the inverse function of f(x) and then substitute x = 2 into the inverse function.
Composite Functions and Inverse Functions
- The composite function g(f(x)) is formed by substituting the entire function f(x) into the function g(x).
- To find the inverse of a composite function, first find the composite function g(f(x)) and then find its inverse using the steps mentioned previously.
- Example:
- g(x) = x² - 2
- f(x) = x + 5
- g(f(x)) = (x + 5)² - 2
- (g(f(x)))⁻¹ = √(x + 2) - 5
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Description
This quiz covers the concept of inverse functions, detailing how they reverse the original function's process. It includes step-by-step instructions for finding inverse functions along with practical examples. Test your understanding of how to evaluate inverse functions at specific values.
