Question

question 3
exponential function evaluation
given the function $f(x) = 990(1.05)^x$ evaluate each of the following.
note: round your answers to two decimal places as needed.
a) evaluate $f(-10)$ $f(-10) = $
b) evaluate $f(-5)$ $f(-5) = $
c) evaluate $f(0)$ $f(0) = $
d) evaluate $f(5)$ $f(5) = $
e) evaluate $f(10)$ $f(10) = $
question help: message instructor

Explanation:

Part A: Evaluate \( f(-10) \)

Step1: Substitute \( x = -10 \) into \( f(x) \)

We have the function \( f(x) = 990(1.05)^x \). Substitute \( x=-10 \) into the function:
\( f(-10)=990\times(1.05)^{-10} \)

Step2: Simplify \( (1.05)^{-10} \)

Recall that \( a^{-n}=\frac{1}{a^{n}} \), so \( (1.05)^{-10}=\frac{1}{(1.05)^{10}} \). Calculate \( (1.05)^{10}\approx1.628894626777 \), then \( \frac{1}{(1.05)^{10}}\approx0.6168389432 \)

Step3: Multiply by 990

\( f(-10)=990\times0.6168389432\approx 990\times0.616839 \approx 605.79 \) (rounded to two decimal places)

Part B: Evaluate \( f(-5) \)

Step1: Substitute \( x = -5 \) into \( f(x) \)

\( f(-5)=990\times(1.05)^{-5} \)

Step2: Simplify \( (1.05)^{-5} \)

Using \( a^{-n}=\frac{1}{a^{n}} \), \( (1.05)^{-5}=\frac{1}{(1.05)^{5}} \). Calculate \( (1.05)^{5}\approx1.2762815625 \), then \( \frac{1}{(1.05)^{5}}\approx0.7835261665 \)

Step3: Multiply by 990

\( f(-5)=990\times0.7835261665\approx 990\times0.783526 \approx 752.43 \) (rounded to two decimal places)

Part C: Evaluate \( f(0) \)

Step1: Substitute \( x = 0 \) into \( f(x) \)

\( f(0)=990\times(1.05)^{0} \)

Step2: Simplify \( (1.05)^{0} \)

Recall that any non - zero number to the power of 0 is 1, so \( (1.05)^{0} = 1 \)

Step3: Multiply by 990

\( f(0)=990\times1 = 990.00 \)

Part D: Evaluate \( f(5) \)

Answer:

\( f(-10) \approx 605.79 \)