EXAMPLE \( \mathbf{3}(\mathbf{2 0 1 8}) \) Given: \( \quad f(x)=\frac{x-3}{x+2} \) 3.1 \( \quad \) Show that \( f(x)=1-\frac{5}{x+2} \) 3.2 \( \quad \) Write down the equations of the vertical and horizontal asymptotes of \( f \). 3.3 \( \quad \) Determine the intercepts of the graph of \( f \) with the \( x \)-axes and \( y \)-axes. \( 3.4 \quad \) Write down the value of \( c \) if \( y=x+c \) is a line of symmetry to the graph of \( f \).

Let's go through each part of the problem step by step. ### 3.1 Show that \( f(x) = 1 - \frac{5}{x+2} \) We start with the given function: \[ f(x) = \frac{x-3}{x+2} \] To show that \( f(x) = 1 - \frac{5}{x+2} \), we can manipulate the right-hand side: 1. Rewrite \( 1 \) with a common denominator: \[ 1 = \frac{x+2}{x+2} \] 2. Now, substitute this into the expression: \[ 1 - \frac{5}{x+2} = \frac{x+2}{x+2} - \frac{5}{x+2} \] 3. Combine the fractions: \[ 1 - \frac{5}{x+2} = \frac{x + 2 - 5}{x + 2} = \frac{x - 3}{x + 2} \] Thus, we have shown that: \[ f(x) = 1 - \frac{5}{x+2} \] ### 3.2 Write down the equations of the vertical and horizontal asymptotes of \( f \) **Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero. Set the denominator equal to zero: \[ x + 2 = 0 \implies x = -2 \] So, the equation of the vertical asymptote is: \[ x = -2 \] **Horizontal Asymptote:** To find the horizontal asymptote, we look at the behavior of \( f(x) \) as \( x \) approaches infinity. The leading coefficients of the numerator and denominator are both 1, so: \[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x-3}{x+2} = \frac{1}{1} = 1 \] Thus, the equation of the horizontal asymptote is: \[ y = 1 \] ### 3.3 Determine the intercepts of the graph of \( f \) **x-intercept:** To find the x-intercept, set \( f(x) = 0 \): \[ \frac{x-3}{x+2} = 0 \implies x - 3 = 0 \implies x = 3 \] So, the x-intercept is: \[ (3, 0) \] **y-intercept:** To find the y-intercept, evaluate \( f(0) \): \[ f(0) = \frac{0-3}{0+2} = \frac{-3}{2} \] So, the y-intercept is: \[ (0, -\frac{3}{2}) \] ### 3.4 Write down the value of \( c \) if \( y = x + c \) is a line of symmetry to the graph of \( f \) For a function to have a line of symmetry of the form \( y = x + c \), the function must be symmetric about this line. This typically occurs when the function can be expressed in a form that reflects this symmetry. To find \( c \), we can use the fact that the line of symmetry \( y = x + c \) implies that for every point \( (a, b) \) on the graph of \( f \), the point \( (b - c, a + c) \) should also be on the graph of \( f \). However, a simpler approach is to recognize that the function \( f(x) \) is a rational function and does not exhibit symmetry about any line of the form \( y = x + c \). To find \( c \), we can use the fact that the horizontal asymptote is \( y = 1 \). The line of symmetry would be \( y = x + c \) where \( c \) is the vertical distance from the horizontal asymptote to the line of symmetry. Since the horizontal asymptote is \( y = 1 \), we can set \( c = -1 \) to have the line of symmetry \( y = x - 1 \). Thus, the value of \( c \) is: \[ c = -1 \]