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Which function has only one [tex]$x$[/tex]-intercept at [tex]$(-6,0)$[/tex]?

A. [tex]$f(x) = x(x-6)$[/tex]
B. [tex]$f(x) = (x-6)(x-6)$[/tex]
C. [tex]$f(x) = (x+6)(x-6)$[/tex]
D. [tex]$f(x) = (x+6)(x+6)$[/tex]

Sagot :

GinnyAnswer
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].
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