To determine which function has only one [tex]\( x \)[/tex]-intercept at [tex]\((-6, 0)\)[/tex], we need to analyze each function:
1. Function: [tex]\( f(x) = x(x-6) \)[/tex]
- Roots: To find the roots, set [tex]\( f(x) = 0 \)[/tex].
[tex]\[
x(x - 6) = 0
\][/tex]
This equation has roots [tex]\( x = 0 \)[/tex] and [tex]\( x = 6 \)[/tex].
- Intercepts: The intercepts are at [tex]\( (0, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex].
- Conclusion: Two intercepts, neither of which is [tex]\((-6, 0)\)[/tex].
2. Function: [tex]\( f(x) = (x-6)(x-6) \)[/tex]
- Roots: To find the roots, set [tex]\( f(x) = 0 \)[/tex].
[tex]\[
(x - 6)(x - 6) = 0
\][/tex]
This equation has a repeated root [tex]\( x = 6 \)[/tex].
- Intercepts: The intercept is at [tex]\( (6, 0) \)[/tex].
- Conclusion: One intercept at [tex]\( (6, 0) \)[/tex], not [tex]\((-6, 0)\)[/tex].
3. Function: [tex]\( f(x) = (x+6)(x-6) \)[/tex]
- Roots: To find the roots, set [tex]\( f(x) = 0 \)[/tex].
[tex]\[
(x + 6)(x - 6) = 0
\][/tex]
This equation has roots [tex]\( x = -6 \)[/tex] and [tex]\( x = 6 \)[/tex].
- Intercepts: The intercepts are at [tex]\( (-6, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex].
- Conclusion: This function has an intercept at [tex]\( (-6, 0) \)[/tex], but also has another intercept at [tex]\( (6, 0) \)[/tex].
4. Function: [tex]\( f(x) = (x+6)(x+6) \)[/tex]
- Roots: To find the roots, set [tex]\( f(x) = 0 \)[/tex].
[tex]\[
(x + 6)(x + 6) = 0
\][/tex]
This equation has a repeated root [tex]\( x = -6 \)[/tex].
- Intercepts: The intercept is at [tex]\( (-6, 0) \)[/tex].
- Conclusion: Only one intercept at [tex]\( (-6, 0) \)[/tex].
By analyzing the intercepts of each function, we conclude that:
The function [tex]\( f(x) = (x+6)(x+6) \)[/tex] has only one [tex]\( x \)[/tex]-intercept at [tex]\((-6, 0)\)[/tex].
