To determine which statement accurately describes the graph of the function [tex]\( g(x) = \frac{1}{5} x \)[/tex] in comparison to the function [tex]\( f(x) = x \)[/tex], let us break down the problem step by step.
### Step 1: Understand the Functions
1. [tex]\( f(x) = x \)[/tex]: This is a linear function where the slope (or steepness) is [tex]\( 1 \)[/tex]. The slope is the coefficient of [tex]\( x \)[/tex], which tells us how steep the line is.
2. [tex]\( g(x) = \frac{1}{5} x \)[/tex]: This is also a linear function, but the slope is [tex]\( \frac{1}{5} \)[/tex].
### Step 2: Compare the Slopes
- The slope of [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex].
- The slope of [tex]\( g(x) \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
### Step 3: Interpretation of Slopes
- If we compare the slopes of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- The slope of [tex]\( f(x) \)[/tex] (which is [tex]\( 1 \)[/tex]) is steeper than the slope of [tex]\( g(x) \)[/tex] (which is [tex]\( \frac{1}{5} \)[/tex]).
- To describe how much steeper [tex]\( f(x) \)[/tex] is compared to [tex]\( g(x) \)[/tex], we see that the slope of [tex]\( g(x) \)[/tex] is [tex]\( \frac{1}{5} \)[/tex] of the slope of [tex]\( f(x) \)[/tex].
### Step 4: Conclusion
By interpreting the comparison of slopes, we can conclude that:
- [tex]\( g(x) \)[/tex] has a slope that is [tex]\( \frac{1}{5} \)[/tex] as steep as the slope of [tex]\( f(x) \)[/tex].
Thus, the correct statement is:
A. The graph of [tex]\( g \)[/tex] is one-fifth as steep as the graph of [tex]\( f \)[/tex].
The numerical result [tex]\( 0.2 \)[/tex] validates this conclusion, as [tex]\( \frac{1}{5} = 0.2 \)[/tex], confirming that the slope of [tex]\( g(x) \)[/tex] is indeed one-fifth of the slope of [tex]\( f(x) \)[/tex].
