Question

answer the questions about the following function. f(x) = (x + 5)/(x - 3) (a) is the point (5,3) on the graph of f? (b) if x = 1, 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. (a) choose the correct answer. a. no, because f(5)≠3. b. yes, because f(5)=3. c. no, because f(3)≠5. d. yes, because f(3)=5. (b) if x = 1, f(x)= - 3. (simplify your answer.) using the information in the previous step, list the point(s) on the graph of f where x = 1. (simplify your answer. type an ordered pair. use a comma to separate answers as needed.)

Explanation:

Step1: Check if point (5,3) is on graph

Substitute x = 5 into f(x)=$\frac{x + 5}{x - 3}$, we get f(5)=$\frac{5+5}{5 - 3}=\frac{10}{2}=5
eq3$.

Step2: Find f(x) when x = 1

Substitute x = 1 into f(x)=$\frac{x + 5}{x - 3}$, f(1)=$\frac{1+5}{1 - 3}=\frac{6}{-2}=-3$. The point on the graph is (1,-3).

Step3: Solve for x when f(x)=2

Set $\frac{x + 5}{x - 3}=2$. Cross - multiply: x + 5 = 2(x - 3). Expand: x + 5 = 2x-6. Rearrange: 2x-x=5 + 6, so x = 11. The point on the graph is (11,2).

Step4: Find domain of f

The denominator of f(x) cannot be 0. Set x - 3=0, x = 3. So the domain is all real numbers except x = 3, or $(-\infty,3)\cup(3,\infty)$.

Step5: Find x - intercepts

Set f(x)=0, so $\frac{x + 5}{x - 3}=0$. The numerator must be 0, x+5 = 0, x=-5. The x - intercept is - 5.

Step6: Find y - intercept

Set x = 0, f(0)=$\frac{0 + 5}{0 - 3}=-\frac{5}{3}$. The y - intercept is $-\frac{5}{3}$.

Answer:

(a) A. No, because f(5)≠3.
(b) (1,-3)
(c) x = 11, (11,2)
(d) $(-\infty,3)\cup(3,\infty)$
(e) - 5
(f) $-\frac{5}{3}$