Question

answer the questions about the following function. f(x) = \frac{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)=□. (simplify your answer.)

Explanation:

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

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

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 is (11,2).

Step4: Find the domain

The denominator of f(x) cannot be 0. So $x-3
eq0$, then $x
eq3$. The domain is $\{x|x
eq3\}$.

Step5: Find x - intercepts

Set f(x)=0. $\frac{x + 5}{x - 3}=0$, then $x+5 = 0$ (since a fraction is 0 when the numerator is 0 and denominator is non - zero), so $x=-5$.

Step6: Find y - intercepts

Set x = 0. $f(0)=\frac{0 + 5}{0 - 3}=-\frac{5}{3}$.

Answer:

(a) A. No, because f(5)≠3.
(b) - 3
(c) 11, the point is (11,2)
(d) $\{x|x
eq3\}$
(e) x=-5
(f) $y =-\frac{5}{3}$