1Cademy - Finding the Domain and Points on the Graph of $$f(x) = \frac{x-1}{x^2-6x+5}$$ when $$f(x) = 4$$

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Strategy to Solve Equations with Rational Expressions

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Finding the Domain and Points on the Graph of $f(x) = \frac{x-1}{x^2-6x+5}$ when $f(x) = 4$

To analyze the rational function $f(x) = \frac{x-1}{x^2-6x+5}$ , we find its domain, solve the equation $f(x) = 4$ , and identify the corresponding point on the graph.

Step 1: Find the domain. Set the denominator to zero to identify undefined values: $x^2 - 6x + 5 = 0$ $(x - 1)(x - 5) = 0$ $x - 1 = 0 \quad \text{or} \quad x - 5 = 0$ $x = 1 \quad \text{or} \quad x = 5$ The domain is all real numbers except $x \neq 1$ and $x \neq 5$ .

Step 2: Solve $f(x) = 4$ . Set the rational expression equal to $4$ : $\frac{x-1}{x^2-6x+5} = 4$ Factor the denominator to find the LCD, which is $(x-1)(x-5)$ : $\frac{x-1}{(x-1)(x-5)} = 4$ Multiply both sides by the LCD to eliminate the fraction: $(x-1)(x-5) \cdot \frac{x-1}{(x-1)(x-5)} = 4(x-1)(x-5)$ Simplify the equation: $x - 1 = 4(x^2 - 6x + 5)$ $x - 1 = 4x^2 - 24x + 20$ Set the equation to zero by subtracting $x$ and adding $1$ to both sides: $0 = 4x^2 - 25x + 21$ Factor the resulting quadratic equation: $0 = (4x - 21)(x - 1)$ Solve for $x$ by setting each factor to zero: $4x - 21 = 0 \quad \text{or} \quad x - 1 = 0$ $x = \frac{21}{4} \quad \text{or} \quad x = 1$ Check for extraneous solutions. The value $x = 1$ is a restricted value from the domain, so it must be discarded. The only valid solution is $x = \frac{21}{4}$ .

Step 3: Find the points on the graph. When $x = \frac{21}{4}$ , the function value is $f(x) = 4$ . The point $\left(\frac{21}{4}, 4\right)$ lies on the graph of the function.

Updated 2026-05-26

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OpenStax Intermediate Algebra

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