Graph the rational function f(x)=(3x-1)/(x-5)

To graph the rational function f(x) = (3x - 1) / (x - 5), follow these steps. Find the vertical asymptote(s) by setting the denominator equal to zero and solving for x.

x - 5 = 0

x = 5

The vertical asymptote of the function is x = 5.

Step 1: Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials. Since the degree of the numerator (3x - 1) is 1, and the degree of the denominator (x - 5) is also 1, the horizontal asymptote is given by the ratio of the leading coefficients:

Horizontal asymptote: y = 3/1

Horizontal asymptote: y = 3

Step 2: Find the x-intercept by setting the numerator equal to zero and solving for x:

3x - 1 = 0

3x = 1

x = 1/3

The x-intercept is (1/3, 0).

Step 3: Find the y-intercept by setting x to zero and evaluating the function:

f(0) = (3 * 0 - 1) / (0 - 5)

f(0) = -1 / -5

f(0) = 1/5

The y-intercept is (0, 1/5).

Step 4: Find additional points to sketch the graph. Choose some x-values around the vertical asymptote (x = 5) and calculate the corresponding y-values. For example, you can use x = 4, x = 6, and x = 8.

x = 4:

f(4) = (3 * 4 - 1) / (4 - 5)

= 11 / -1

= -11

x = 6:

f(6) = (3 * 6 - 1) / (6 - 5)

= 17 / 1

= 17

x = 8:

f(8) = (3 * 8 - 1) / (8 - 5)

= 23 / 3

≈ 7.67

Step 5: Plot the points obtained from Steps 2 to 5 and draw the graph. The graph should approach the horizontal asymptote (y = 3) as it approaches the vertical asymptote (x = 5).

The graph of the rational function f(x) = (3x - 1) / (x - 5) should look like this:

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