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].
