To solve this problem, we need to determine the [tex]\( x \)[/tex]-intercept and [tex]\( y \)[/tex]-intercept of the linear function [tex]\( f(x) = 4x + 12 \)[/tex].
### 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. At this point, the value of [tex]\( f(x) \)[/tex] is 0. Thus, we set the equation equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = 4x + 12
\][/tex]
Subtract 12 from both sides:
[tex]\[
-12 = 4x
\][/tex]
Divide both sides by 4:
[tex]\[
x = -3
\][/tex]
Hence, the [tex]\( x \)[/tex]-intercept is [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. At this point, the value of [tex]\( x \)[/tex] is 0. We substitute [tex]\( x \)[/tex] with 0 in the function:
[tex]\[
f(0) = 4(0) + 12 = 12
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, 12)\)[/tex].
### Conclusion:
Based on our calculations, we find that:
- The [tex]\( x \)[/tex]-intercept is [tex]\((-3, 0)\)[/tex]
- The [tex]\( y \)[/tex]-intercept is [tex]\((0, 12)\)[/tex]
Therefore, 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].
