To find the [tex]\( x \)[/tex]-intercept and [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) = 4x + 12 \)[/tex], let's go through the steps in detail.
### Finding the [tex]\( x \)[/tex]-Intercept
The [tex]\( x \)[/tex]-intercept is the point where the graph of the function crosses the [tex]\( x \)[/tex]-axis. This occurs when [tex]\( f(x) = 0 \)[/tex].
1. Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
0 = 4x + 12
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
4x + 12 = 0
\][/tex]
[tex]\[
4x = -12
\][/tex]
[tex]\[
x = \frac{-12}{4}
\][/tex]
[tex]\[
x = -3
\][/tex]
So, the [tex]\( x \)[/tex]-intercept is at the point [tex]\((-3, 0)\)[/tex].
### Finding the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept is the point where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
1. Evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 4 \cdot 0 + 12
\][/tex]
2. Calculate the value:
[tex]\[
f(0) = 12
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is at the point [tex]\((0, 12)\)[/tex].
Based on this detailed solution, the correct answer is:
D. The [tex]\( x \)[/tex]-intercept is [tex]\((-3,0)\)[/tex], and the [tex]\( y \)[/tex]-intercept is [tex]\((0, 12)\)[/tex].
