Question
Let \( f(x)=\frac{1}{x} \) Write a formula for a function \( g \) whose graph is obtained from the graph of \( f(x) \) after the following sequence of transformations: (1) scale vertically by a factor of 5 ; (2) shift left 5 unit (3) shift up 3 units. \( g(x)= \)
Ask by Rowe Evans. in the United States
Jan 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( g(x) = \frac{5}{x + 5} + 3 \)
Solution
To construct the function \( g(x) \) based on the given transformations applied to \( f(x) = \frac{1}{x} \), follow these steps in the specified order:
1. **Scale Vertically by a Factor of 5:**
- This transformation multiplies the original function by 5.
\[
5f(x) = 5 \cdot \frac{1}{x} = \frac{5}{x}
\]
2. **Shift Left by 5 Units:**
- Shifting the function left by 5 units involves replacing \( x \) with \( x + 5 \).
\[
5f(x + 5) = 5 \cdot \frac{1}{x + 5} = \frac{5}{x + 5}
\]
3. **Shift Up by 3 Units:**
- Shifting the function up by 3 units adds 3 to the entire function.
\[
g(x) = \frac{5}{x + 5} + 3
\]
**Final Formula:**
\[
g(x) = \frac{5}{x + 5} + 3
\]
**Answer:**
After simplifying, the function g is 5 ⁄ (x + 5) plus three. Thus,
g(x) = 5/(x + 5) + 3
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Simplify this solution Bonus Knowledge
To transform the function \( f(x) = \frac{1}{x} \) through the specified sequence of transformations: 1. **Scale vertically by a factor of 5**: This changes \( f(x) \) to \( 5f(x) \). \[ g(x) = 5 \cdot \frac{1}{x} = \frac{5}{x} \] 2. **Shift left 5 units**: This means we replace \( x \) with \( x + 5 \). \[ g(x) = \frac{5}{x + 5} \] 3. **Shift up 3 units**: Adding 3 to the function gives us: \[ g(x) = \frac{5}{x + 5} + 3 \] Putting it all together, we arrive at the final formula for \( g(x) \): \[ g(x) = \frac{5}{x + 5} + 3 \]
