Question
What are the \( x \)-intercepts of the given rational function below? \( f(x)=\frac{x-4}{x^{2}+2 x-48} \) \( (5,0) \) and \( (1,0) \) \( (4,0) \) \( (0,0) \) \( (1,0) \)
Ask by Powell Daniel. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The \( x \)-intercept is at \( (4, 0) \).
Solution
To find the \( x \)-intercepts of the rational function \( f(x) = \frac{x - 4}{x^{2} + 2x - 48} \), we need to determine the values of \( x \) that make \( f(x) = 0 \).
### Steps to Find the \( x \)-Intercepts:
1. **Set the numerator equal to zero:**
\[
x - 4 = 0 \implies x = 4
\]
2. **Ensure the denominator is not zero at \( x = 4 \):**
\[
x^2 + 2x - 48 = (4)^2 + 2(4) - 48 = 16 + 8 - 48 = -24 \neq 0
\]
Since the denominator is not zero when \( x = 4 \), this is a valid \( x \)-intercept.
3. **Conclusion:**
The function has a single \( x \)-intercept at \( (4, 0) \).
### Answer:
\( (4,\ 0) \)
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Simplify this solution The Deep Dive
To find the \( x \)-intercepts of the rational function \( f(x)=\frac{x-4}{x^{2}+2x-48} \), you need to set the numerator equal to zero since \( f(x) = 0 \) when the numerator is zero (and the denominator is not). Here, the numerator is \( x - 4 \), which equals zero when \( x = 4 \). Therefore, the \( x \)-intercept is \( (4,0) \). Now, watch out for common mix-ups! A common mistake is thinking that the denominator can also be zero when finding \( x \)-intercepts. Remember, you only care about the numerator being zero, while the denominator must not equal zero for valid intercepts.
