Question
Given: \( f(x)=\frac{3 x-2}{x^{2}+10 x+25} \) Determine the values of \( x \) for which \( f(x) \) is defined
Ask by Li Macdonald. in South Africa
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
f(x) is defined for all real numbers except x = -5.
Solution
To determine the values of x for which the function is defined, we need to ensure that the denominator is not equal to zero.
The denominator is:
x² + 10x + 25
We can factor this quadratic expression:
x² + 10x + 25 = (x + 5)²
The denominator is zero when (x + 5)² = 0, which gives:
x + 5 = 0
x = -5
Thus, the function f(x) is undefined at x = -5. For all other values of x, the function is defined.
Answer: f(x) is defined for all real numbers except x = -5.
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Simplify this solution Beyond the Answer
To determine the values of \( x \) for which the function \( f(x) = \frac{3x - 2}{x^2 + 10x + 25} \) is defined, we need to identify where the denominator does not equal zero. The denominator can be factored as \( x^2 + 10x + 25 = (x + 5)^2 \). This expression is equal to zero when \( x + 5 = 0 \), which means \( x = -5 \). Therefore, \( f(x) \) is undefined at \( x = -5 \). In conclusion, \( f(x) \) is defined for all real numbers except \( x = -5 \).
