Question
Sketch the inverse graph of f(x)=2x for the domain -5 less than equal to x is less than 5?
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Search instead for Skecth th inverse graph of f(x)=2^x for the domain -5 less than equal to x is less than 5?
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Answered by
GPT-4o mini
To find the inverse graph of a function, we follow the basic steps of swapping the \(x\) and \(y\) variables in the original function and then solving for \(y\).
### 1. Inverse of \( f(x) = 2x \)
1. **Original function**:
\[ y = 2x \]
2. **Swap \(x\) and \(y\)**:
\[ x = 2y \]
3. **Solve for \(y\)**:
\[ y = \frac{x}{2} \]
So, the inverse function is \( f^{-1}(x) = \frac{x}{2} \).
### Graphing the inverse of \( f(x) = 2x \)
- The original function \( f(x) = 2x \) is a straight line passing through the origin with a slope of 2.
- The inverse function \( f^{-1}(x) = \frac{x}{2} \) is also a straight line passing through the origin, but with a slope of \( \frac{1}{2} \).
**Domain and Range**: Both the function and its inverse will have the same domain and range:
- Domain of \( f(x) = 2x \): \([-5, 5]\)
- Range of \( f(x) = 2x \): \([-10, 10]\)
- Domain of \( f^{-1}(x) = \frac{x}{2} \): \([-10, 10]\)
- Range of \( f^{-1}(x) = \frac{x}{2} \): \([-5, 5]\)
### 2. Inverse of \( f(x) = 2^x \)
1. **Original function**:
\[ y = 2^x \]
2. **Swap \(x\) and \(y\)**:
\[ x = 2^y \]
3. **Solve for \(y\)**:
To isolate \(y\), take the logarithm (base 2) of both sides:
\[ y = \log_2(x) \]
So, the inverse function is \( f^{-1}(x) = \log_2(x) \).
### Graphing the inverse of \( f(x) = 2^x \)
- The original function \( f(x) = 2^x \) is an exponential growth curve, which, for the specified domain of \([-5, 5]\), ranges from approximately \(0.03125\) (for \(x = -5\)) to \(32\) (for \(x = 5\)).
- The inverse function \( f^{-1}(x) = \log_2(x) \) is a logarithmic function.
**Domain and Range**:
- Domain of \( f(x) = 2^x \): \([-5, 5]\)
- Range of \( f(x) = 2^x \): \([0.03125, 32]\)
- Domain of \( f^{-1}(x) = \log_2(x) \): \((0, 32]\)
- Range of \( f^{-1}(x) = \log_2(x) \): \([-5, 5]\)
### Summary
To sketch the inverse graphs:
- For \( f(x) = 2x \) and \( f^{-1}(x) = \frac{x}{2} \):
- Both are linear functions, where the inverse is less steep.
- For \( f(x) = 2^x \) and \( f^{-1}(x) = \log_2(x) \):
- \( f(x) \) is an exponential curve rising steeply as \(x\) increases, while \( f^{-1}(x) \) is a logarithmic curve that increases gradually over its domain.
If you're sketching these functions, plot points, and consider the symmetry across the line \( y = x \) which reflects the function and its inverse.
### 1. Inverse of \( f(x) = 2x \)
1. **Original function**:
\[ y = 2x \]
2. **Swap \(x\) and \(y\)**:
\[ x = 2y \]
3. **Solve for \(y\)**:
\[ y = \frac{x}{2} \]
So, the inverse function is \( f^{-1}(x) = \frac{x}{2} \).
### Graphing the inverse of \( f(x) = 2x \)
- The original function \( f(x) = 2x \) is a straight line passing through the origin with a slope of 2.
- The inverse function \( f^{-1}(x) = \frac{x}{2} \) is also a straight line passing through the origin, but with a slope of \( \frac{1}{2} \).
**Domain and Range**: Both the function and its inverse will have the same domain and range:
- Domain of \( f(x) = 2x \): \([-5, 5]\)
- Range of \( f(x) = 2x \): \([-10, 10]\)
- Domain of \( f^{-1}(x) = \frac{x}{2} \): \([-10, 10]\)
- Range of \( f^{-1}(x) = \frac{x}{2} \): \([-5, 5]\)
### 2. Inverse of \( f(x) = 2^x \)
1. **Original function**:
\[ y = 2^x \]
2. **Swap \(x\) and \(y\)**:
\[ x = 2^y \]
3. **Solve for \(y\)**:
To isolate \(y\), take the logarithm (base 2) of both sides:
\[ y = \log_2(x) \]
So, the inverse function is \( f^{-1}(x) = \log_2(x) \).
### Graphing the inverse of \( f(x) = 2^x \)
- The original function \( f(x) = 2^x \) is an exponential growth curve, which, for the specified domain of \([-5, 5]\), ranges from approximately \(0.03125\) (for \(x = -5\)) to \(32\) (for \(x = 5\)).
- The inverse function \( f^{-1}(x) = \log_2(x) \) is a logarithmic function.
**Domain and Range**:
- Domain of \( f(x) = 2^x \): \([-5, 5]\)
- Range of \( f(x) = 2^x \): \([0.03125, 32]\)
- Domain of \( f^{-1}(x) = \log_2(x) \): \((0, 32]\)
- Range of \( f^{-1}(x) = \log_2(x) \): \([-5, 5]\)
### Summary
To sketch the inverse graphs:
- For \( f(x) = 2x \) and \( f^{-1}(x) = \frac{x}{2} \):
- Both are linear functions, where the inverse is less steep.
- For \( f(x) = 2^x \) and \( f^{-1}(x) = \log_2(x) \):
- \( f(x) \) is an exponential curve rising steeply as \(x\) increases, while \( f^{-1}(x) \) is a logarithmic curve that increases gradually over its domain.
If you're sketching these functions, plot points, and consider the symmetry across the line \( y = x \) which reflects the function and its inverse.