Solve for domain, range, x-interept and y-intercept.[tex]f(x) = \frac{x + 6}{x - 2} [/tex]
Answer (1)
Domain, range, x and y intercepts[tex]$\sf{f(x) = \frac{x+6}{x-2}}$[/tex][tex]\\[/tex]1. DomainThe domain is the set of all possible x values for which the function is defined.Rational functions are undefined where the denominator is zero.Set the denominator equal to zero and solve for x:[tex]$\sf{x - 2 = 0}$[/tex][tex]$\sf{x=2}$[/tex]Therefore, the domain is all real numbers except x = 2.Domain: [tex]$\sf{(-\infty, 2) \cup (2, \infty)}[/tex][tex]\\[/tex]2. RangeThe range is the set of all possible y values (or f(x) values) that the function can take.To find the range, we can analyze the horizontal asymptote of the rational function.Since the degree of the numerator and denominator are the same (both are 1), the horizontal asymptote is the ratio of the leading coefficients: [tex]$\sf{y = \frac{1}{1} = 1}$[/tex].This means the function approaches y = 1 as x goes to positive or negative infinity, but it never actually reaches 1.Range: [tex]$\sf{(-\infty, 1) \cup (1, \infty)}$[/tex][tex]\\[/tex]3. x-interceptThe x-intercept is the point where the graph crosses the x-axis, which means f(x) = 0. [tex]$\sf{0 = \frac{x+6}{x-2}}$[/tex][tex]$\sf{0 = x + 6}$[/tex][tex]$\sf{x = -6}$[/tex]x-intercept: (-6, 0)[tex]\\[/tex]4. y-interceptThe y-intercept is the point where the graph crosses the y-axis, which means x = 0.[tex]$\sf{f(0) = \frac{0+6}{0-2} = \frac{6}{-2} = -3}$[/tex]y-intercept: (0, -3)[tex]\\[/tex]Final answers (summary):Domain: [tex]$\sf{(-\infty, 2) \cup (2, \infty)}$[/tex]Range: [tex]$\sf{(-\infty, 1) \cup (1, \infty)}$[/tex]x-intercept: [tex]$\sf{(-6, 0)}$[/tex]y-intercept: [tex]$\sf{(0, -3)}$[/tex]
