To determine which function has two [tex]\(x\)[/tex]-intercepts, one at [tex]\((0,0)\)[/tex] and one at [tex]\((4,0)\)[/tex], we need to examine the given functions to find their intercepts.
### 1) [tex]\( f(x) = x(x - 4) \)[/tex]
First, set [tex]\( f(x) = 0 \)[/tex] to find the [tex]\(x\)[/tex]-intercepts:
[tex]\[ x(x - 4) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = 0 \quad \text{or} \quad x - 4 = 0 \][/tex]
[tex]\[ x = 0 \quad \text{or} \quad x = 4 \][/tex]
These intercepts are [tex]\((0, 0)\)[/tex] and [tex]\((4, 0)\)[/tex].
### 2) [tex]\( f(x) = x(x + 4) \)[/tex]
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x(x + 4) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x = 0 \quad \text{or} \quad x + 4 = 0 \][/tex]
[tex]\[ x = 0 \quad \text{or} \quad x = -4 \][/tex]
These intercepts are [tex]\((0, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex].
### 3) [tex]\( f(x) = (x - 4)(x - 4) \)[/tex]
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ (x - 4)(x - 4) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
This intercept is [tex]\((4, 0)\)[/tex], but it is a repeated root, so there's only one unique intercept at [tex]\((4, 0)\)[/tex].
### 4) [tex]\( f(x) = (x + 4)(x + 4) \)[/tex]
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ (x + 4)(x + 4) = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we get:
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4 \][/tex]
This intercept is [tex]\((-4, 0)\)[/tex], but it is a repeated root, so there's only one unique intercept at [tex]\((-4, 0)\)[/tex].
### Conclusion
From the analysis above, the function [tex]\( f(x) = x(x - 4) \)[/tex] has [tex]\( x \)[/tex]-intercepts at [tex]\((0,0)\)[/tex] and [tex]\((4,0)\)[/tex].
Thus, the correct function is:
[tex]\[ \boxed{f(x) = x(x - 4)} \][/tex]
