$f(x)=\frac{x-3}{x^2+9 x-22}$
Find $f(2)$. If the value of the function is undefined write "no solutions". If the value of the function is complex write "no solutions".
Answer (1)
The problem asks to evaluate the function f ( x ) = x 2 + 9 x − 22 x − 3 at x = 2 .
Substitute x = 2 into the function.
Calculate the numerator: 2 − 3 = − 1 .
Calculate the denominator: 2 2 + 9 ( 2 ) − 22 = 0 .
Since the denominator is zero, the function is undefined, so the answer is no solutions .
Explanation
Problem Analysis We are given the function f ( x ) = x 2 + 9 x − 22 x − 3 and asked to find f ( 2 ) . We need to substitute x = 2 into the function and evaluate it. If the denominator is zero, the function is undefined. If the result is a complex number, we write 'no solutions'.
Substitution First, we substitute x = 2 into the function: f ( 2 ) = 2 2 + 9 ( 2 ) − 22 2 − 3
Calculate Numerator Next, we calculate the numerator: 2 − 3 = − 1
Calculate Denominator Then, we calculate the denominator: 2 2 + 9 ( 2 ) − 22 = 4 + 18 − 22 = 22 − 22 = 0
Conclusion Since the denominator is zero, the function is undefined at x = 2 . Therefore, the answer is 'no solutions'.
Examples
In electrical engineering, the transfer function of a circuit can sometimes be represented as a rational function similar to the one in this problem. Determining the value of the function at a specific frequency (analogous to x) helps engineers understand the circuit's behavior at that frequency. If the denominator becomes zero, it indicates a resonance condition, which can have significant implications for the circuit's design and performance.
