Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Certainly! Here's the transcribed text tailored for an educational website:
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### Mathematics Exercise: Inverse Functions
**Problem Statement:**
1. Given the function \( f(x) = \frac{x^3}{2} + 4 \) with the domain \( 1 \leq x \leq 3 \):
a) Find the formula for the inverse function \( f^{-1}(x) \).
b) Determine the domain of \( f^{-1}(x) \).
---
For visual aids accompanying this problem, consider diagrams illustrating the function \( f(x) \) and its inverse \( f^{-1}(x) \). These diagrams can include:
- **Graph of \( f(x) \):** A cubic graph opening to the right, showing the range for \( x \) within [1, 3].
- **Graph of \( f^{-1}(x) \):** A reflection of the graph of \( f(x) \) over the line \( y = x \), illustrating the domain and range swap for the inverse function.
These visualizations can help in understanding the transformation from the original function to its inverse.
---](https://content.bartleby.com/qna-images/question/dcae3a3d-b28c-4acd-9ee5-e15685d9d008/89eee61d-3d8a-477a-b04a-ef3c21f2ba22/8az84z.jpeg)
Transcribed Image Text:Certainly! Here's the transcribed text tailored for an educational website:
---
### Mathematics Exercise: Inverse Functions
**Problem Statement:**
1. Given the function \( f(x) = \frac{x^3}{2} + 4 \) with the domain \( 1 \leq x \leq 3 \):
a) Find the formula for the inverse function \( f^{-1}(x) \).
b) Determine the domain of \( f^{-1}(x) \).
---
For visual aids accompanying this problem, consider diagrams illustrating the function \( f(x) \) and its inverse \( f^{-1}(x) \). These diagrams can include:
- **Graph of \( f(x) \):** A cubic graph opening to the right, showing the range for \( x \) within [1, 3].
- **Graph of \( f^{-1}(x) \):** A reflection of the graph of \( f(x) \) over the line \( y = x \), illustrating the domain and range swap for the inverse function.
These visualizations can help in understanding the transformation from the original function to its inverse.
---
![**Problem Statement:**
Write an equation of the quadratic function with the vertex at the point \((-1, 4)\) and such that \(f(0) = 6\).
---
**Solution:**
To find the equation of a quadratic function given its vertex, we use the vertex form of a quadratic equation:
\[ f(x) = a(x - h)^2 + k \]
Where \( (h, k) \) is the vertex of the parabola. Given the vertex \((-1, 4)\), the equation becomes:
\[ f(x) = a(x + 1)^2 + 4 \]
We also know that \( f(0) = 6 \). Substitute \( x = 0 \) into the equation to find \( a \):
\[ 6 = a(0 + 1)^2 + 4 \]
\[ 6 = a(1) + 4 \]
\[ 2 = a \]
Thus, the equation of the quadratic function is:
\[ f(x) = 2(x + 1)^2 + 4 \]](https://content.bartleby.com/qna-images/question/dcae3a3d-b28c-4acd-9ee5-e15685d9d008/89eee61d-3d8a-477a-b04a-ef3c21f2ba22/1kjxf4g_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Write an equation of the quadratic function with the vertex at the point \((-1, 4)\) and such that \(f(0) = 6\).
---
**Solution:**
To find the equation of a quadratic function given its vertex, we use the vertex form of a quadratic equation:
\[ f(x) = a(x - h)^2 + k \]
Where \( (h, k) \) is the vertex of the parabola. Given the vertex \((-1, 4)\), the equation becomes:
\[ f(x) = a(x + 1)^2 + 4 \]
We also know that \( f(0) = 6 \). Substitute \( x = 0 \) into the equation to find \( a \):
\[ 6 = a(0 + 1)^2 + 4 \]
\[ 6 = a(1) + 4 \]
\[ 2 = a \]
Thus, the equation of the quadratic function is:
\[ f(x) = 2(x + 1)^2 + 4 \]
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