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Problem 7. a) Describe the transformation(s) that can be applied to the function g(x) = e* to obtain the graph of f(x) = e¨*+7. b) State the domain and range of f(x). c) Sketch the graph of f(x).

Problem 7. a) Describe the transformation(s) that can be applied to the function g(x) = e* to obtain the graph of f(x) = e¨*+7. b) State the domain and range of f(x). c) Sketch the graph of f(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 7
### Problem 7 #### a) Describe the transformation(s) that can be applied to the function \( g(x) = e^x \) to obtain the graph of \( f(x) = e^{-x} + 7 \). To obtain the graph of \( f(x) = e^{-x} + 7 \) from \( g(x) = e^x \), you can apply the following transformations: 1. **Reflection over the y-axis**: \( e^x \) becomes \( e^{-x} \). 2. **Vertical shift upwards**: \( e^{-x} \) is shifted up by 7 units, resulting in \( e^{-x} + 7 \). #### b) State the domain and range of \( f(x) \). - **Domain**: The domain of \( f(x) = e^{-x} + 7 \) is all real numbers \( \mathbb{R} \). In mathematical notation: \( (-\infty, \infty) \). - **Range**: The range of \( f(x) = e^{-x} + 7 \) is \( (7, \infty) \), since \( e^{-x} \) is always positive and approaches 0 as \( x \) tends to \( \infty \), thus \( e^{-x} + 7 \) will always be greater than 7. #### c) Sketch the graph of \( f(x) \). To sketch the graph of \( f(x) = e^{-x} + 7 \): 1. **Start with the basic graph of \( g(x) = e^x \)**: This is an exponentially increasing function passing through the point (0,1) and asymptotic to the x-axis as \( x \) approaches \( -\infty \). 2. **Apply a reflection over the y-axis**: The graph of \( e^{-x} \) is a reflection of \( e^x \) around the y-axis, which means it will start high on the left, decreasing as \( x \) increases. 3. **Shift the graph vertically upwards by 7 units**: To go from \( e^{-x} \) to \( e^{-x} + 7 \), raise the entire graph up by 7. The new horizontal asymptote will be the line \( y = 7 \). The graph will pass through
Transcribed Image Text:### Problem 7 #### a) Describe the transformation(s) that can be applied to the function \( g(x) = e^x \) to obtain the graph of \( f(x) = e^{-x} + 7 \). To obtain the graph of \( f(x) = e^{-x} + 7 \) from \( g(x) = e^x \), you can apply the following transformations: 1. **Reflection over the y-axis**: \( e^x \) becomes \( e^{-x} \). 2. **Vertical shift upwards**: \( e^{-x} \) is shifted up by 7 units, resulting in \( e^{-x} + 7 \). #### b) State the domain and range of \( f(x) \). - **Domain**: The domain of \( f(x) = e^{-x} + 7 \) is all real numbers \( \mathbb{R} \). In mathematical notation: \( (-\infty, \infty) \). - **Range**: The range of \( f(x) = e^{-x} + 7 \) is \( (7, \infty) \), since \( e^{-x} \) is always positive and approaches 0 as \( x \) tends to \( \infty \), thus \( e^{-x} + 7 \) will always be greater than 7. #### c) Sketch the graph of \( f(x) \). To sketch the graph of \( f(x) = e^{-x} + 7 \): 1. **Start with the basic graph of \( g(x) = e^x \)**: This is an exponentially increasing function passing through the point (0,1) and asymptotic to the x-axis as \( x \) approaches \( -\infty \). 2. **Apply a reflection over the y-axis**: The graph of \( e^{-x} \) is a reflection of \( e^x \) around the y-axis, which means it will start high on the left, decreasing as \( x \) increases. 3. **Shift the graph vertically upwards by 7 units**: To go from \( e^{-x} \) to \( e^{-x} + 7 \), raise the entire graph up by 7. The new horizontal asymptote will be the line \( y = 7 \). The graph will pass through
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