Question

1. the graph of $y = 2^x - 4$ is positive on which interval? (1) $(-infty,infty)$ (3) $(0,infty)$ (2) $(2,infty)$ (4) $(-4,infty)$ 17. the asymptote of the graph of $f(x) = 5log(x + 4)$ is (1) $y = 6$ (3) $x = 4$ (2) $x = -4$ (4) $y = 5$ 18. a research assistant receives a first year salary of $90,000 and a 2% annual raise throughout the first ten years of employment. in total, how much money will be earned over the first ten years, to the nearest dollar? (1) $91,837$ (3) $877,917$ (2) $109,709$ (4) $985,475$

Explanation:

Question 16

Step1: Set $y>0$

$2^x - 4 > 0$

Step2: Isolate exponential term

$2^x > 4$

Step3: Rewrite 4 as $2^2$

$2^x > 2^2$

Step4: Solve for $x$ (since $2^x$ is increasing)

$x > 2$

Step1: Identify log domain restriction

For $\log(x+4)$, $x+4 > 0$

Step2: Find vertical asymptote

$x+4 = 0 \implies x = -4$

Step1: Recognize geometric series

This is a finite geometric series where $a_1=90000$, $r=1.02$, $n=10$.

Step2: Use sum formula

$S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$

Step3: Substitute values

$S_{10} = 90000 \cdot \frac{(1.02)^{10} - 1}{1.02 - 1}$

Step4: Calculate $(1.02)^{10}$

$(1.02)^{10} \approx 1.2190$

Step5: Compute numerator/denominator

$\frac{1.2190 - 1}{0.02} = \frac{0.2190}{0.02} = 10.95$

Step6: Find total salary

$90000 \times 10.95 = 985500$ (matches rounded value)

Answer:

(2) $(2,\infty)$

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Question 17