What is the domain of the function $f(x)=\frac{x+1}{x^2-6 x+8}$?
A. all real numbers
B. all real numbers except -1
C. all real numbers except -4 and -2
D. all real numbers except 2 and 4
Answer (2)
Find the values of x for which the denominator x 2 − 6 x + 8 equals zero.
Factor the quadratic equation: x 2 − 6 x + 8 = ( x − 2 ) ( x − 4 ) = 0 .
Solve for x : x = 2 and x = 4 .
The domain of the function is all real numbers except 2 and 4: all real numbers except 2 and 4 .
Explanation
Analyze the problem We are given the function f ( x ) = x 2 − 6 x + 8 x + 1 and asked to find its domain. The domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero.
Set up the equation To find the values of x that make the denominator zero, we need to solve the quadratic equation x 2 − 6 x + 8 = 0 .
Factor the quadratic We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 8 and add to -6. These numbers are -2 and -4. So, we can factor the quadratic as ( x − 2 ) ( x − 4 ) = 0 .
Solve for x Setting each factor equal to zero gives us the solutions x − 2 = 0 and x − 4 = 0 . Solving for x , we get x = 2 and x = 4 .
State the domain Therefore, the domain of the function f ( x ) is all real numbers except x = 2 and x = 4 .
Examples
Consider designing a bridge where the load distribution is modeled by a rational function. The domain restrictions (where the denominator is zero) represent critical points where the structure could become unstable. Understanding the domain ensures the bridge's stability by avoiding these critical points, ensuring safe load management and structural integrity.
The domain of the function f ( x ) = x 2 − 6 x + 8 x + 1 is all real numbers except for x = 2 and x = 4 . Therefore, the answer is all real numbers except 2 and 4 .
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