Question

Answer the questions about the following function.
x + 6
f(x) =
x-12
(a) Is the point (5,-2) on the graph of f?
(b) If x= 3, what is f(x)? What point is on the graph of f?
(c) If f(x) = 2, what is x? What point(s) is (are) on the graph of f?
(d) What is the domain of f?
(e) List the x-intercepts, if any, of the graph of f.
(f) List the y-intercept, if there is one, of the graph of f.
(g) What are the zeros of f?

Answer 1

The given rational function has a vertical asymptote at x = 12, and the

graph has x and y-intercept at the negative x and y-axis.

Response:

(a) The point (5, -2) is not on the graph

(b) f(x) = -1

• The point on the graph of f is (-1, 3)

(c) If f(x) = 2, x = 30

• The point on the graph of f is (30, 2)

(d) Domain = (-∞, 12) ∪ (12, ∞)

(e) x-intercept is at x = -6, which is the point (-6, 0)

(f) The y-intercept is at y = -0.5, which is the point (0, -0.5)

(g) A zeros of f, is x = -6

Which method is used to evaluate the rational function?

The possible given function is presented as follows;

$f(x) =\mathbf{\dfrac{x + 6}{x - 12}}$

(a) When x = -2, we have;

$f(-2) = \dfrac{(-2) + 6}{(-2) - 12} = \dfrac{4}{-14} = \mathbf{-\dfrac{2}{7}}$

Therefore;

The point ($-\frac{2}{7}$, -2) is on the graph, which gives;

• The point (5, -2) is not on the graph

(b) If x = 3, we have;

$f(3) =\mathbf{ \dfrac{3 + 6}{3 - 12}} = \dfrac{9}{-9} =-1$

• f(3) = -1

• The point on the graph of f is (-1, 3)

(c)  If f(x) = 2, we have;

$f(x) =2 = \mathbf{ \dfrac{x + 6}{x - 12}}$

2·(x - 12) = x + 6

2·x - x = 6 + 2 × 12 = 30

x = 30

• If f(x) = 2, x = 30

• The point on the graph of f is (30, 2)

(d) The  domain which is the list of possible x-values, which is expressed as follows;

Domain; -∞ < x < 12, and 12 < x < ∞

Which gives;

• Domain = (-∞, 12) ∪ (12, ∞)

(e) The x-intercepts are;

$0 = \mathbf{ \dfrac{x + 6}{x - 12}}$

• x-intercept is at x = -6, which is (-6, 0)

(f) The y-intercept

$At \ the \ y-intercept, \ f(x) = \dfrac{0 + 6}{0 - 12} = -\dfrac{1}{2} = \mathbf{ -0.5}$

• The y-intercept is at y = -0.5, which is (0, -0.5)

(g) The zeros of the graph, f, are the points at which the graph crosses the x-axis which at the point x = -6

• A zeros of the graph is x = -6

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