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Section 6-1 : Exponential Functions

2. Given the function \(f\left( x \right) = {\left( {\displaystyle \frac{1}{5}} \right)^x}\) evaluate each of the following.

  1. \(f\left( { - 3} \right)\)
  2. \(f\left( { - 1} \right)\)
  3. \(f\left( 0 \right)\)
  4. \(f\left( 2 \right)\)
  5. \(f\left( 3 \right)\)

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a \(f\left( { - 3} \right)\) Show Solution

All we need to do here is plug in the \(x\) and do any quick arithmetic we need to do.

\[f\left( { - 3} \right) = {\left( {\frac{1}{5}} \right)^{ - 3}} = {\left( {\frac{5}{1}} \right)^3} = \frac{{{5^3}}}{{{1^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{125}}\]

b \(f\left( { - 1} \right)\) Show Solution

All we need to do here is plug in the \(x\) and do any quick arithmetic we need to do.

\[f\left( - \right) = {\left( {\frac{1}{5}} \right)^{ - \,1}} = {\left( {\frac{5}{1}} \right)^1} = \require{bbox} \bbox[2pt,border:1px solid black]{5}\]

c \(f\left( 0 \right)\) Show Solution

All we need to do here is plug in the \(x\) and do any quick arithmetic we need to do.

\[f\left( 0 \right) = {\left( {\frac{1}{5}} \right)^0} = \require{bbox} \bbox[2pt,border:1px solid black]{1}\]

d \(f\left( 2 \right)\) Show Solution

All we need to do here is plug in the \(x\) and do any quick arithmetic we need to do.

\[f\left( 2 \right) = {\left( {\frac{1}{5}} \right)^2} = \frac{{{1^2}}}{{{5^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{25}}}}\]

e \(f\left( 3 \right)\) Show Solution

All we need to do here is plug in the \(x\) and do any quick arithmetic we need to do.

\[f\left( 3 \right) = {\left( {\frac{1}{5}} \right)^3} = \frac{{{1^3}}}{{{5^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{125}}}}\]