To determine which function has two [tex]$x$[/tex]-intercepts, one at [tex]$(0,0)$[/tex] and one at [tex]$(4,0)$[/tex], let's analyze each given function step by step.
### 1. [tex]\( f(x) = x(x-4) \)[/tex]
- Intercept at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0 \cdot (0 - 4) = 0
\][/tex]
So, [tex]\((0, 0)\)[/tex] is an intercept.
- Intercept at [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 4 \cdot (4 - 4) = 0
\][/tex]
So, [tex]\((4, 0)\)[/tex] is also an intercept.
This function has the required intercepts at [tex]\((0,0)\)[/tex] and [tex]\((4,0)\)[/tex].
### 2. [tex]\( f(x) = x(x+4) \)[/tex]
- Intercept at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 0 \cdot (0 + 4) = 0
\][/tex]
So, [tex]\((0, 0)\)[/tex] is an intercept.
- Intercept at [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = (-4) \cdot (-4 + 4) = 0
\][/tex]
So, [tex]\((-4, 0)\)[/tex] is an intercept, but we need [tex]\((4, 0)\)[/tex].
This function does not fulfill the requirement for the second intercept at [tex]\((4,0)\)[/tex].
### 3. [tex]\( f(x) = (x-4)(x-4) \)[/tex]
- Intercept at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = (0 - 4)(0 - 4) = 16 \neq 0
\][/tex]
So, [tex]\((0, 0)\)[/tex] is not an intercept.
- Intercept at [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = (4 - 4)(4 - 4) = 0
\][/tex]
So, [tex]\((4, 0)\)[/tex] is an intercept.
This function does not have [tex]\((0, 0)\)[/tex] as an intercept, only [tex]\((4, 0)\)[/tex].
### 4. [tex]\( f(x) = (x+4)(x+4) \)[/tex]
- Intercept at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = (0 + 4)(0 + 4) = 16 \neq 0
\][/tex]
So, [tex]\((0, 0)\)[/tex] is not an intercept.
- Intercept at [tex]\( x = -4 \)[/tex]:
[tex]\[
f(-4) = (-4 + 4)(-4 + 4) = 0
\][/tex]
So, [tex]\((-4, 0)\)[/tex] is an intercept, not [tex]\((4, 0)\)[/tex].
This function does not fulfill the requirement for the intercepts at [tex]\((0, 0)\)[/tex] and [tex]\((4, 0)\)[/tex].
### Conclusion:
From our analysis, the correct function that has [tex]$x$[/tex]-intercepts at both [tex]$(0,0)$[/tex] and [tex]$(4,0)$[/tex] is:
[tex]\[ f(x) = x(x-4) \][/tex]
So, the correct answer is the first option, [tex]\( f(x) = x(x-4) \)[/tex].
