4.1 The equation of the asymptotes of 𝑓(𝑥) = 3𝑥−2 + 1 is x = 2/3 and y = 1. This is because the vertical asymptote of a rational function is found by setting the denominator equal to zero and solving for x, and the horizontal asymptote can be found by looking at the degrees of the numerator and denominator.

4.2 The x-intercepts of 𝑓(𝑥) are found by setting the function equal to zero and solving for x. So, 0 = 3𝑥−2 + 1. Solving this equation gives x = 2/3.

4.3 The y-intercepts of 𝑓(𝑥) are found by setting x = 0 in the function. So, 𝑓(0) = 3(0)−2 + 1 = 1.

4.4 To sketch the graph of 𝑓 and ℎ on the same set of axes, plot the points and asymptotes found in the previous steps. The graph of 𝑓 will approach the asymptotes but never cross them. The graph of ℎ(𝑥) =− 1𝑥+524 will be a straight line with slope -1 and y-intercept 524.

4.5 To find the values for which 𝑓(𝑥)

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4.1 The equation of the asymptotes of 𝑓(𝑥) = 3𝑥−2 + 1 is x = 2/3 and y = 1. This is because the vertical asymptote of a rational function is found by setting the denominator equal to zero and solving for x, and the horizontal asymptote can be found by looking at the degrees of the numerator and denominator.

4.2 The x-intercepts of 𝑓(𝑥) are found by setting the function equal to zero and solving for x. So, 0 = 3𝑥−2 + 1. Solving this equation gives x = 2/3.

4.3 The y-intercepts of 𝑓(𝑥) are found by setting x = 0 in the function. So, 𝑓(0) = 3(0)−2 + 1 = 1.

4.4 To sketch the graph of 𝑓 and ℎ on the same set of axes, plot the points and asymptotes found in the previous steps. The graph of 𝑓 will approach the asymptotes but never cross them. The graph of ℎ(𝑥) =− 1𝑥+524 will be a straight line with slope -1 and y-intercept 524.

4.5 To find the values for which 𝑓(𝑥) < ℎ(𝑥), set the two functions equal to each other and solve for x: 3𝑥−2 + 1 = -1𝑥+524. Solve this equation to find the x values.

4.6 The line 𝑦 =− 34 𝑥 + 254 cuts 𝑓 at 𝐸(3; 4). To find the coordinates of 𝐹, set 𝑦 =− 34 𝑥 + 254 equal to 𝑓(𝑥) and solve for x. Then substitute this x value into either the equation of the line or the function to find the corresponding y value.

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4.1 The equation of the asymptotes of 𝑓(𝑥) = 3𝑥−2 + 1 is x = 2/3 and y = 1. This is because the vertical asymptote of a rational function is found by setting the denominator equal to zero and solving for x, and the horizontal asymptote can be found by looking at the degrees of the numerator and denominator.

4.2 The x-intercepts of 𝑓(𝑥) are found by setting the function equal to zero and solving for x. So, 0 = 3𝑥−2 + 1. Solving this equation gives x = 2/3.

4.3 The y-intercepts of 𝑓(𝑥) are found by setting x = 0 in the function. So, 𝑓(0) = 3(0)−2 + 1 = 1.

4.4 To sketch the graph of 𝑓 and ℎ on the same set of axes, plot the points and asymptotes found in the previous steps. The graph of 𝑓 will approach the asymptotes but never cross them. The graph of ℎ(𝑥) =− 1𝑥+524 will be a straight line with slope -1 and y-intercept 524.

4.5 To find the values for which 𝑓(𝑥) < ℎ(𝑥), set the two functions equal to each other and solve for x: 3𝑥−2 + 1 = -1𝑥+524. Solve this equation to find the x values.

4.6 The line 𝑦 =− 34 𝑥 + 254 cuts 𝑓 at 𝐸(3; 4). To find the coordinates of 𝐹, set 𝑦 =− 34 𝑥 + 254 equal to 𝑓(𝑥) and solve for x. Then substitute this x value into either the equation of the line or the function to find the corresponding y value.

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