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Math  /  Algebra

QuestionThe function ff is defined by the following rule. f(x)=(110)xf(x)=\left(\frac{1}{10}\right)^{x}
Find f(x)f(x) for each xx-value in the table. \begin{tabular}{|c|c|} \hlinexx & f(x)f(x) \\ \hline-1 & □\square \\ \hline 0 & □\square \\ \hline 1 & □\square \\ \hline 2 & □\square \\ \hline \end{tabular}

Studdy Solution

STEP 1

1. The function is defined as f(x)=(110)x f(x) = \left(\frac{1}{10}\right)^x .
2. We need to evaluate f(x) f(x) for specific values of x x given in the table.

STEP 2

1. Evaluate f(x) f(x) for x=−1 x = -1 .
2. Evaluate f(x) f(x) for x=0 x = 0 .
3. Evaluate f(x) f(x) for x=1 x = 1 .
4. Evaluate f(x) f(x) for x=2 x = 2 .

STEP 3

Evaluate f(x) f(x) for x=−1 x = -1 :
f(−1)=(110)−1 f(-1) = \left(\frac{1}{10}\right)^{-1}

STEP 4

Simplify the expression using the property of exponents (1a)−n=an\left(\frac{1}{a}\right)^{-n} = a^n:
f(−1)=101=10 f(-1) = 10^1 = 10

STEP 5

Evaluate f(x) f(x) for x=0 x = 0 :
f(0)=(110)0 f(0) = \left(\frac{1}{10}\right)^0

STEP 6

Simplify the expression using the property that any non-zero number raised to the power of 0 is 1:
f(0)=1 f(0) = 1

STEP 7

Evaluate f(x) f(x) for x=1 x = 1 :
f(1)=(110)1 f(1) = \left(\frac{1}{10}\right)^1

STEP 8

Simplify the expression:
f(1)=110 f(1) = \frac{1}{10}

STEP 9

Evaluate f(x) f(x) for x=2 x = 2 :
f(2)=(110)2 f(2) = \left(\frac{1}{10}\right)^2

STEP 10

Simplify the expression:
f(2)=110×110=1100 f(2) = \frac{1}{10} \times \frac{1}{10} = \frac{1}{100}
The values of f(x) f(x) for each x x in the table are:
\begin{tabular}{|c|c|} \hline x & f(x) \\ \hline -1 & 10 \\ \hline 0 & 1 \\ \hline 1 & \frac{1}{10} \\ \hline 2 & \frac{1}{100} \\ \hline \end{tabular}

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