Firstly, we need to determine the steepness of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. The steepness of a function is evaluated by calculating the slope of the function.
### 1. Slope of [tex]\( f(x) \)[/tex]
To find the slope of [tex]\( f(x) \)[/tex], we can use the values given in the table. We use the first and last points in the dataset to find the slope:
[tex]\[ x_1 = -9, \; f(x_1) = 176 \][/tex]
[tex]\[ x_2 = 9, \; f(x_2) = -148 \][/tex]
The slope [tex]\( m_f \)[/tex] of [tex]\( f(x) \)[/tex] is given by the formula:
[tex]\[
m_f = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Plugging in the values, we get:
[tex]\[
m_f = \frac{-148 - 176}{9 - (-9)}
\][/tex]
[tex]\[
m_f = \frac{-148 - 176}{9 + 9}
\][/tex]
[tex]\[
m_f = \frac{-324}{18}
\][/tex]
[tex]\[
m_f = -18
\][/tex]
### 2. Slope of [tex]\( g(x) \)[/tex]
The function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = 14 - 23x \][/tex]
The slope [tex]\( m_g \)[/tex] of [tex]\( g(x) \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is:
[tex]\[
m_g = -23
\][/tex]
### 3. Comparison of Slopes
Now we compare the absolute values of the slopes to determine which function is steeper:
[tex]\[
|m_f| = 18
\][/tex]
[tex]\[
|m_g| = 23
\][/tex]
Since [tex]\(|m_g| > |m_f|\)[/tex], the graph of [tex]\( g(x) \)[/tex] is steeper than the graph of [tex]\( f(x) \)[/tex].
### Conclusion
Based on this comparison, we determine which statement is true.
- A. The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same steepness with negative slopes. - Incorrect because [tex]\(|m_g| \neq |m_f|\)[/tex].
- B. The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are both horizontal lines. - Incorrect because neither slope is zero.
- C. The graph of [tex]\( g(x) \)[/tex] is steeper than the graph of [tex]\( f(x) \)[/tex]. - Correct since [tex]\(|m_g| > |m_f|\)[/tex].
- D. The graph of [tex]\( g(x) \)[/tex] is less steep than the graph of [tex]\( f(x) \)[/tex]. - Incorrect because [tex]\(|m_g| < |m_f|\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\text{C}}
\][/tex]
