If f(x)=10xf(x)=10xand g=a⋅f(x)g=a⋅f(x)for some a∈Ra∈R, which of the following is only attribute that will change for **ANY** a≠1a≠1?

The only attribute that will change for **any** \( a \neq 1 \) in the function \( g(x) = a \cdot f(x) \), where \( f(x) = 10x \), is the **slope** of the line.

### Explanation:

1. **Given Functions:**
   - \( f(x) = 10x \)
   - \( g(x) = a \cdot f(x) = 10ax \)

2. **Analysis of Attributes:**
   - The function \( f(x) = 10x \) is a linear function with a slope of 10 and a y-intercept of 0.
   - The function \( g(x) = 10ax \) is also a linear function, but its slope is \( 10a \) and its y-intercept remains 0.

3. **Effect of \( a \):**
   - **Slope:** The slope of \( g(x) \) is \( 10a \). Therefore, the slope changes whenever \( a \) changes from 1.
   - **Y-intercept:** Since \( g(x) \) is simply a scaled version of \( f(x) \) without any added constant, the y-intercept remains at 0 for all \( a \).

### Conclusion:
The **slope** is the only attribute that will change for **any** \( a \neq 1 \).

Would you like any further details or have any questions?

Here are 5 related questions you might consider:
1. How does the graph of \( g(x) \) compare to \( f(x) \) when \( a > 1 \) or \( a < 1 \)?
2. What would be the effect of adding a constant to \( f(x) \) on the y-intercept of \( g(x) \)?
3. How would the slope of \( g(x) \) change if \( f(x) \) were a quadratic function instead of linear?
4. What happens to the graph of \( g(x) \) when \( a = -1 \)?
5. Can you express the relationship between \( a \) and the steepness of \( g(x) \) compared to \( f(x) \)?

**Tip:** Always check both the slope and y-intercept when analyzing how linear functions transform under different operations.

Explore how linear functions and slope change with different values of a in g(x) = a * f(x), where f(x) = 10x. Understand the impact on function graphs and key attributes such as slope and y-intercept. Suitable for high school students studying functions and linear algebra.

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