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 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 \(\frac{x + 5}{x-3}=2\), then \(x + 5=2(x - 3)\). Expand to get \(x+5 = 2x-6\). Rearrange terms: \(2x-x=5 + 6\), so \(x = 11\). The point on the graph is \((11,2)\).

Step4: Find domain of \(f\)

The function \(f(x)=\frac{x + 5}{x-3}\) is undefined when the denominator is 0. Set \(x-3=0\), \(x = 3\). So the domain is all real numbers except \(x = 3\), written as \((-\infty,3)\cup(3,\infty)\).

Step5: Find \(x\)-intercepts

Set \(y = f(x)=0\), then \(\frac{x + 5}{x-3}=0\). A fraction is 0 when the numerator is 0 and denominator is non - zero. Set \(x+5=0\), \(x=-5\). The \(x\) - intercept is \(-5\).

Step6: Find \(y\)-intercept

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

Answer:

(a) A. No, because \(f(5)
eq3\).
(b) \(-3\), the point \((1,-3)\) is on the graph of \(f\).
(c) \(x = 11\), the point \((11,2)\) is on the graph of \(f\).
(d) \((-\infty,3)\cup(3,\infty)\)
(e) \(-5\)
(f) \(-\frac{5}{3}\)