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

Given the functions \( f(x) = 10x \) and \( g(x) = a \cdot f(x) \) where \( a \in \mathbb{R} \), we need to determine which attribute of the function will change for any \( a \neq 1 \).

### Step 1: Understand the functions
- \( f(x) = 10x \) is a linear function with a slope of 10.
- \( g(x) = a \cdot f(x) = a \cdot 10x = 10ax \) is also a linear function, but with a slope of \( 10a \).

### Step 2: Consider the effect of \( a \)
The function \( g(x) \) is essentially a scaling of the function \( f(x) \) by the factor \( a \). Here’s what changes:
- **Slope:** The slope of \( f(x) \) is 10. For \( g(x) \), the slope becomes \( 10a \). Therefore, the slope changes when \( a \) changes, provided \( a \neq 1 \).

### Step 3: Attributes of linear functions
Linear functions like \( f(x) \) and \( g(x) \) have two primary attributes:
1. **Slope**: Determines the steepness of the line.
2. **Y-intercept**: The value where the function crosses the y-axis.

Since \( g(x) \) is a scaling of \( f(x) \) and does not involve any vertical translation, the **only attribute that will change** for any \( a \neq 1 \) is the **slope**.

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

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Would you like any further details or have any questions?

Here are 5 related questions you might consider:
1. What happens to the graph of \( g(x) \) when \( a < 0 \)?
2. How does the y-intercept change if \( f(x) = 10x + b \) instead of \( 10x \)?
3. What is the effect of setting \( a = 0 \) on \( g(x) \)?
4. How would the derivative of \( g(x) \) change with different values of \( a \)?
5. Can you describe the transformation of the graph of \( f(x) \) when \( a > 1 \)?

**Tip:** When analyzing linear functions, always pay attention to how the slope and y-intercept are affected by any transformations.

Explore how scaling affects linear functions using examples of functions like f(x) = 10x and g(x) = a ⋅ f(x). Understand how the slope of these functions changes with different scaling factors, focusing on mathematical concepts of linearity and slope. Suitable for high school students and anyone interested in algebraic transformations.

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