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Reed College Department of Mathematics & Statistics

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Worksheet Exponential Functions

1.

Find the formula for an exponential equation that passes through the points, (0,4) and (1,5). The exponential equation should be of the form \(y=ab^x\text{.}\)
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If the video does not work, click here to watch on YouTube.
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Answer.
\(y = 4\cdot 1.25^{x}\)
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2.

For the function \(f(x)= 9(7)^x\text{,}\) find \(f(3)\text{.}\) Enter answer as a simplified exact value.
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\(f(3)=\)
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Answer.
\(3087\)
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3.

For the function \(f(x)= 2.1(2)^{-x+1}+1.3\text{,}\) find \(f(5)\text{.}\) Round answer to four decimal places, if necessary.
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\(f(5)=\)
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Answer.
\(1.4313\)
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4.

For the exponential function \(f(x)=\) `3(3)^(x-2)`, find \(f(7)\text{.}\)
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\(f(7)=\)
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Answer.
\(729\)
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5.

Determine whether the following equation represents exponential growth, exponential decay, or neither.
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\(y=12.8(1.395)^{\frac{1}{x}}\)
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Answer.
\(\text{Neither}\)
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6.

Consider the equation \(y=170(1.23)^x\text{,}\) which is of the form \(y = ab^x\text{,}\) to fill in the following blanks.
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The initial value for equation is . The base, denoted by \(b\text{,}\) for the equation is . Therefore, the type of change represented is
because
.
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Answer 1.
\(170\)
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Answer 2.
\(1.23\)
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Answer 3.
\(\text{exponential growth}\)
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Answer 4.
\({\verb!b>1!}\)
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7.

Consider the equation \(y=426(0.72)^x\text{,}\) which is of the form \(y = ab^x\text{,}\) to fill in the following blanks.
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The initial value for equation is . The base, denoted by \(b\text{,}\) for the equation is . Therefore, the type of change represented is
because
.
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Answer 1.
\(426\)
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Answer 2.
\(0.72\)
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Answer 3.
\(\text{continuous decay}\)
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Answer 4.
\({\verb!0<b<1!}\)
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8.

Determine whether the table could represent a function that is linear, exponential, or neither. If the function is exponential or linear, find a function that passes through the points. If the function is neither exponential nor linear, type NONE.
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\(x\) 1 2 3 4
\(f(x)\) 70 40 10 -20
\(f(x)=\)
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Answer 1.
\(\text{linear}\)
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Answer 2.
\(-30x+100\)
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9.

Determine whether the table could represent a function that is linear, exponential, or neither. If the function is exponential or linear, find a function that passes through the points. If the function is neither, type NONE.
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\(x\) 1 2 3 4
\(f(x)\) 70 49 34.3 24.01
\(f(x) =\)
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Answer 1.
\(\text{exponential}\)
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Answer 2.
\(100\cdot 0.7^{x}\)
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10.

Determine whether the following statements represent an exponential function or a linear function.
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  • The average annual population of a pack of wolves increases each year by 25 wolves.
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  • The value of a coin collection has increased by 3.25% annually over the last 20 years.
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  • A population of bacteria decreases by a factor of \(\frac{1}{4}\) every 24 hours.
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  • For the first 6 training sessions, a personal trainer charges his clients $5 less than the previous training session.
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  • The population of a certain country at time `t` is represented by the function \(f(t)=50000(1.02)^{.25t}\text{.}\)
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Answer 1.
\(\text{linear function}\)
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Answer 2.
\(\text{exponential function}\)
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Answer 3.
\(\text{exponential function}\)
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Answer 4.
\(\text{linear function}\)
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Answer 5.
\(\text{exponential function}\)
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11.

For the following exercises, consider this scenario:
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For each year \(t\text{,}\) the number of trees in Forest A is represented by the function \(A(t)=115({0.97})^t\text{.}\) In the nearby Forest B, the number of trees is represented by the function \(B(t)=96({0.91})^t\text{.}\)
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  1. Which forest’s population is decreasing at a faster rate?
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  2. Which forest had a greater number of trees initially?
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  3. Initially, by how many trees did the larger forest exceed the smaller forest?
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  4. Which forest will have a greater number of trees after 30 years?
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  5. After 29 years, by how many trees will the larger forest exceed the smaller forest? Round to the nearest whole number.
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If the video does not work, click here to watch on YouTube.
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Answer 1.
\(\text{Forest B}\)
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Answer 2.
\(\text{Forest A}\)
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Answer 3.
\(19\)
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Answer 4.
\(\text{Forest A}\)
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Answer 5.
\(41\)
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12.

For the following exercises, consider this scenario:
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For each year \(t\text{,}\) the number of otters in Ocean \(A\) is represented by the function \(A(t)=2058({0.95})^t\text{.}\) In Ocean \(B\) , the number of sea otters is represented by the function \(B(t)=3855({0.83})^t\text{.}\)
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  1. Which oceans’s sea otter population is declining at a faster rate?
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  2. Which ocean had a greater number of sea otters initially?
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  3. Initially, by how many sea otters did Ocean \(B\) exceed Ocean \(A\text{?}\)
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  4. Which ocean will have a lesser number of sea otters after \(\displaystyle{23}\) years?
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  5. After \(\displaystyle{23}\) years, by how many sea otter inhabitants will Ocean \(A\) exceed Ocean \(B\text{?}\) Round to the nearest whole number.
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Answer 1.
\(\text{Ocean B}\)
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Answer 2.
\(\text{Ocean B}\)
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Answer 3.
\(1797\)
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Answer 4.
\(\text{Ocean B}\)
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Answer 5.
\(579\)
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13.

Evaluate the following expression.
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\(2^{2}\) =
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  • Do not use decimal approximations.
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  • Simplify your answer completely.
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Answer.
\(4\)
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Solution.
\(2^{2} \rightarrow (2)(2) \rightarrow {4}\)
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14.

Evaluate the following expression.
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\(\large{ 27^{ \frac{1}{3} } = }\)
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  • Do not use decimal approximations.
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  • Simplify your answer completely.
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Hint.
Remember that rational exponents such as \(\displaystyle B^{1/n}\) refer to the nth root of B.
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Algebraically: \(\displaystyle B^{1/n} = \sqrt[n]{B}\)
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Answer.
\(3\)
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Solution.
\(27^{ \frac{1}{3} } \rightarrow \sqrt[3]{27} \rightarrow {3}\)
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15.

Evaluate the following expression.
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\(3^{-3}\) =
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  • Do not use decimal approximations.
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  • Simplify your answer completely.
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Answer.
\({\frac{1}{27}}\)
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Solution.
\(3^{-3} \rightarrow \left(\dfrac{1}{3}\right)^{3} \rightarrow {{\frac{1}{27}}}\)
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16.

Solve for x.
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\(\large{ 64^{ x } = 4 }\)
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\(x =\)
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  • Do not use decimal approximations.
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  • Simplify your answer completely.
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Hint.
If we start by recognizing that \(4^{3} = 64\text{,}\) that’s good - but we’re not looking for an exponent of 4.
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What power of 64 is needed in order to make 4?
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Answer.
\({\frac{1}{3}}\)
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Solution.
If we start by recognizing that \(4^{3} = 64\text{,}\) that’s good - but we’re not looking for an exponent of 4.
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We must rewrite our equation as \(4 = 64^{\frac{1}{3}}\) because we’re looking for an exponent of 64 in this problem.
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\(x = \frac{1}{3}\) because \(\sqrt[3]{64} = 4\) and \(\sqrt[3]{64} \rightarrow 4^{\frac{1}{3}}\)
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17.

Solve for x.
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\(\large{ 5^{ x } = 125 }\)
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\(x =\)
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  • Do not use decimal approximations.
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  • Simplify your answer completely.
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Hint.
You’re asked to find the exponent of 5 that makes 125.
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Answer.
\(3\)
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Solution.
125 is \(5^{3}\text{,}\) so 3 is the power of 5 that makes 125.
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\(x = 3\)
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19.

Suppose \(f\) is an exponential function. If \(f(2) = 216\) and \(f(5) = 46656\text{,}\) then find \(f(x)\text{.}\)
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  • \(\displaystyle f(x) = 2\cdot 6^{x}\)
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  • \(\displaystyle f(x) = 4\cdot 3^{x}\)
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  • \(\displaystyle f(x) = 6\cdot 4^{x}\)
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  • \(\displaystyle f(x) = 6\cdot 6^{x}\)
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Answer.
\(\text{Choice 4}\)
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