Before graphing a rational function, see if you cans 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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Factorize Numerator and Denominator: Simplify the rational function by factoring if possible. We will first look at the numerator and denominator to see if they can be factored. Numerator: x^2 - 4x + 1 Denominator: 2x - 3 Neither the numerator nor the denominator has common factors, and the numerator does not factor nicely into a product of binomials. Therefore, we cannot simplify the function by factoring.Rewrite Using Division: Rewrite using division and partial fractions if possible. Since the degree of the numerator is less than the degree of the denominator, we cannot perform polynomial long division. Also, partial fractions are not applicable here because the denominator is not a product of linear factors. Therefore, we cannot simplify the function using division or partial fractions.Identify Graph Holes: Find any holes in the graph. A hole occurs in the graph of a rational function when there is a common factor in the numerator and denominator that can be canceled. Since we have already determined that there are no common factors, there are no holes in the graph of this function.Find Vertical Asymptotes: Find any vertical asymptotes. Vertical asymptotes occur where the denominator equals zero, as long as the numerator does not also equal zero at that point (which would indicate a hole instead). Set the denominator equal to zero and solve for x: 2x - 3 = 0 2x = 3 x = 3/2 There is a vertical asymptote at x = 3/2.Find Horizontal Asymptotes: Find any horizontal asymptotes. Horizontal asymptotes are determined by the degrees of the numerator and denominator. Since the degree of the numerator (2) is less than the degree of the denominator (1), there is a horizontal asymptote at y = 0.Determine Domain: Determine the domain of the function. The domain of a rational function is all real numbers except where the denominator is zero. Since the denominator is zero at x = 3/2, the domain is all real numbers except x = 3/2. Domain: x ∈ ℝ, x ≠ 3/2Determine Range: Determine the range of the function. The range of a rational function can be more challenging to determine and often requires graphing the function or considering its behavior. However, because there is a horizontal asymptote at y = 0 and a vertical asymptote at x = 3/2, we can infer that the function takes on all real values except possibly some restrictions around the asymptotes. Without further analysis or graphing, we cannot specify the exact range, but we can say that it will include all real numbers except possibly values close to the horizontal asymptote. Range: y ∈ ℝ, possibly excluding values near y = 0Use Intercepts: Use intercepts as needed. To find the x-intercept, set f(x) to zero and solve for x: 0 = (x^2 - 4x + 1)/(2x - 3) This implies that the numerator must be zero: x^2 - 4x + 1 = 0 This quadratic does not factor easily, so we would use the quadratic formula to find the x-intercepts: x = [4 ± √(16 - 4(1)(1))]/2 x = [4 ± √(12)]/2 x = [4 ± 2√3]/2 x = 2 ± √3 So, the x-intercepts are at x = 2 + √3 and x = 2 - √3.  To find the y-intercept, set x to zero in the function: f(0) = (0^2 - 4(0) + 1)/(2(0) - 3) f(0) = 1/(-3) The y-intercept is at (0, -1/3).

Before graphing a rational function, see if you cans 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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