Join IDNLearn.com to access a wealth of knowledge and get your questions answered by experts. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.

The table represents the function [tex]\( f(x) \)[/tex].

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -9 & -6 & -3 & 0 & 3 & 6 & 9 \\
\hline
[tex]$f(x)$[/tex] & 176 & 122 & 68 & 14 & -40 & -94 & -148 \\
\hline
\end{tabular}

If [tex]\( g(x) = 14 - 23x \)[/tex], which statement is true?

A. The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same steepness with negative slopes.
B. The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are both horizontal lines.
C. The graph of [tex]\( g(x) \)[/tex] is steeper than the graph of [tex]\( f(x) \)[/tex].
D. The graph of [tex]\( g(x) \)[/tex] is less steep than the graph of [tex]\( f(x) \)[/tex].


Sagot :

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]