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

\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 0 & 1 & 4 & 9 & 16 \\
\hline
[tex]$f(x)$[/tex] & -4 & -3 & -2 & -1 & 0 \\
\hline
\end{tabular}

If [tex]$g(x) = 4 \sqrt{x} - 8$[/tex], which statement is true?

A. The [tex]$y$[/tex]-intercept of [tex]$g(x)$[/tex] is less than the [tex]$y$[/tex]-intercept of [tex]$f(x)$[/tex].

B. The [tex]$y$[/tex]-intercept of [tex]$g(x)$[/tex] is equal to the [tex]$y$[/tex]-intercept of [tex]$f(x)$[/tex].

C. The [tex]$x$[/tex]-intercept of [tex]$g(x)$[/tex] is equal to the [tex]$x$[/tex]-intercept of [tex]$f(x)$[/tex].

D. The [tex]$x$[/tex]-intercept of [tex]$g(x)$[/tex] is greater than the [tex]$x$[/tex]-intercept of [tex]$f(x)$[/tex].

Sagot :

GinnyAnsweridn
To determine which statement is true, we need to find the [tex]\( y \)[/tex]-intercept and [tex]\( x \)[/tex]-intercept for both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] and compare them.

1. Finding the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:

The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. From the table, we see:
[tex]\[ f(0) = -4 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-4\)[/tex].

2. Finding the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:

The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4 \sqrt{0} - 8 = 4(0) - 8 = -8 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\(-8\)[/tex].

3. Finding the [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:

The [tex]\( x \)[/tex]-intercept occurs where [tex]\( f(x) = 0 \)[/tex]. From the table, we see:
[tex]\[ f(16) = 0 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( 16 \)[/tex].

4. Finding the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex]:

The [tex]\( x \)[/tex]-intercept occurs where [tex]\( g(x) = 0 \)[/tex]. Set the equation to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = 4 \sqrt{x} - 8 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 4 \sqrt{x} = 8 \][/tex]
[tex]\[ \sqrt{x} = 2 \][/tex]
[tex]\[ x = 4 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 4 \)[/tex].

Next, we compare the intercepts:

- Comparing [tex]\( y \)[/tex]-intercepts: The [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\(-8\)[/tex], and the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\(-4\)[/tex]. Since [tex]\(-8\)[/tex] is less than [tex]\(-4\)[/tex], the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is less than the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex].

- Comparing [tex]\( x \)[/tex]-intercepts: The [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is [tex]\( 4 \)[/tex], and the [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is [tex]\( 16 \)[/tex]. Since [tex]\( 4 \)[/tex] is less than [tex]\( 16 \)[/tex], the [tex]\( x \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is less than the [tex]\( x \)[/tex]-intercept of [tex]\( f(x) \)[/tex].

Therefore, the correct statement is:

A. The [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is less than the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex].
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