Before graphing a rational function, see if you can simplify it by factoring. Rewrite using division and partial fractions if possible. Then find any holes and asymptotes. Use intercepts as needed. What is the domain? Range? f(x) = (x^2 - 4x + 1)/(2x-3)

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Factorization Attempt: Factor the numerator and denominator if possible to simplify the rational function. The numerator is x^2 - 4x + 1 and the denominator is 2x - 3. We will try to factor both.Numerator Factorization Check: Check if the numerator x^2 - 4x + 1 can be factored. The discriminant of the quadratic equation ax^2 + bx + c is b^2 - 4ac. For x^2 - 4x + 1, the discriminant is (-4)^2 - 4(1)(1) = 16 - 4 = 12. Since the discriminant is positive but not a perfect square, the quadratic does not factor over the integers.Denominator Factorization Check: Check if the denominator 2x - 3 can be factored. The denominator is already in its simplest linear form and cannot be factored further.Identify Graph Holes: Identify any holes in the graph. A hole occurs if a factor is common to both the numerator and the denominator and can be canceled out. Since the numerator and denominator have no common factors, there are no holes in the graph.Vertical Asymptote: Find the vertical asymptote(s). A vertical asymptote occurs where the denominator equals zero and the numerator is not zero at that point. Set the denominator 2x - 3 to zero and solve for x: 2x - 3 = 0, x = 3/2. There is a vertical asymptote at x = 3/2.Horizontal Asymptote: Find the horizontal asymptote(s). For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote. Here, the degrees of the numerator and denominator are the same, so the horizontal asymptote is y = 1/2.X-Intercepts: Find the x-intercept(s). To find the x-intercept(s), set the numerator equal to zero and solve for x: x^2 - 4x + 1 = 0. This quadratic equation does not factor nicely, so we use the quadratic formula: x = [4 ± sqrt(12)]/2. The x-intercepts are x = 2 ± sqrt(3).Y-Intercepts: Find the y-intercept(s). To find the y-intercept(s), set x = 0 in the function and solve for y: f(0) = (0^2 - 4*0 + 1)/(2*0 - 3) = 1/(-3). The y-intercept is y = -1/3.Domain Determination: Determine the domain of the function. The domain of a function is the set of all possible x-values. For this rational function, the only restriction is the vertical asymptote at x = 3/2, where the function is undefined. Therefore, the domain is all real numbers except x = 3/2.Range Determination: Determine the range of the function. The range of a function is the set of all possible y-values. For this rational function, there are no restrictions on the y-values, so the range is all real numbers.

Before graphing a rational function, see if you can simplify it by factoring. Rewrite using division and partial fractions if possible. Then find any holes and asymptotes. Use intercepts as needed. What is the domain? Range? $f(x) = \frac{x^2 - 4x + 1}{2x-3}$

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