Chapter10: Exponential And Logarithmic Functions
Section10.1: Finding Composite And Inverse Functions
Problem 64E: Explain how to find the inverse of a function from its equation. Use an example to demonstrate the...
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![### Inverse Function Calculation Example
Consider the function defined as follows:
\[ f(x) = 2x + 14 \]
We are tasked with finding the inverse of this function, denoted by \( f^{-1}(x) \). In order to find the inverse, we need to perform the following steps:
1. **Start with the original equation:**
\[ y = 2x + 14 \]
2. **Swap \( x \) and \( y \) to begin solving for the inverse:**
\[ x = 2y + 14 \]
3. **Solve this new equation for \( y \):**
\[ x - 14 = 2y \]
\[ y = \frac{x - 14}{2} \]
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{x - 14}{2} \]
#### Input Field for Verification
You may input the calculated inverse function in the provided input field and use the "Preview" button to verify your answer.
\[ f^{-1}(x) = \ \boxed{\ \ \ \ \ \ \ \ \ \ } \]
<button disabled>Preview</button>
By following these steps, you can find the inverse function for any given linear function. This process helps to understand the relationship between the original function and its inverse, which essentially reverses the effect of the original function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F244f2dd3-56b0-4e55-8cec-63e02ab2a541%2Ff8599e45-c4b4-4b9b-bb15-f1c1bbc83adf%2Fj88vfw7.png&w=3840&q=75)
Transcribed Image Text:### Inverse Function Calculation Example
Consider the function defined as follows:
\[ f(x) = 2x + 14 \]
We are tasked with finding the inverse of this function, denoted by \( f^{-1}(x) \). In order to find the inverse, we need to perform the following steps:
1. **Start with the original equation:**
\[ y = 2x + 14 \]
2. **Swap \( x \) and \( y \) to begin solving for the inverse:**
\[ x = 2y + 14 \]
3. **Solve this new equation for \( y \):**
\[ x - 14 = 2y \]
\[ y = \frac{x - 14}{2} \]
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{x - 14}{2} \]
#### Input Field for Verification
You may input the calculated inverse function in the provided input field and use the "Preview" button to verify your answer.
\[ f^{-1}(x) = \ \boxed{\ \ \ \ \ \ \ \ \ \ } \]
<button disabled>Preview</button>
By following these steps, you can find the inverse function for any given linear function. This process helps to understand the relationship between the original function and its inverse, which essentially reverses the effect of the original function.
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