To determine the [tex]\( x \)[/tex]-intercept and the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) = 4x + 12 \)[/tex], we follow these steps:
1. Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 4 \cdot 0 + 12 = 12
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, 12)\)[/tex].
2. Finding the [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept occurs where the graph of the function crosses the [tex]\( x \)[/tex]-axis. This happens when [tex]\( f(x) = 0 \)[/tex].
- Set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
4x + 12 = 0
\][/tex]
- Subtract 12 from both sides:
[tex]\[
4x = -12
\][/tex]
- Divide both sides by 4:
[tex]\[
x = -3
\][/tex]
So, the [tex]\( x \)[/tex]-intercept is [tex]\((-3, 0)\)[/tex].
Therefore, the [tex]\( x \)[/tex]-intercept is [tex]\((-3, 0)\)[/tex] and the [tex]\( y \)[/tex]-intercept is [tex]\((0, 12)\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{\text{D. The } x\text{-intercept is }(-3,0)\text{, and the }y\text{-intercept is }(0,12)\text{.}} \][/tex]
