Solved on Nov 27, 2023

Find the values of $g(x) = (\frac{1}{4})^x$ for $x = -2, -1, 0, 1, 2$ .

STEP 1

Assumptions1. The function $g$ is defined as $g(x) = \left(\frac{1}{4}\right)^{x}$ . . We need to find the values of $g(x)$ for $x = -, -1,0,1,$ .

STEP 2

Let's start with $x = -2$ . We need to substitute $-2$ for $x$ in the function $g(x)$ .
$g(-2) = \left(\frac{1}{4}\right)^{-2}$

STEP 3

The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . So, we can rewrite $\left(\frac{1}{}\right)^{-2}$ as $\frac{1}{\left(\frac{1}{}\right)^2}$ .
$g(-2) = \frac{1}{\left(\frac{1}{}\right)^2}$

STEP 4

Calculate the value of $\left(\frac{1}{4}\right)^2$ .
$\left(\frac{1}{4}\right)^2 = \frac{1}{16}$

STEP 5

Substitute the value of $\left(\frac{1}{4}\right)^2$ back into the equation.
$g(-2) = \frac{1}{\frac{1}{16}}$

STEP 6

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{1}{\frac{1}{16}}$ is equivalent to $1 \times16$ .
$g(-2) =1 \times16$

STEP 7

Calculate the value of $1 \times16$ .
$g(-2) =16$

STEP 8

Now, let's find $g(-1)$ by substituting $-1$ for $x$ in the function $g(x)$ .
$g(-1) = \left(\frac{1}{4}\right)^{-1}$

STEP 9

Rewrite $\left(\frac{}{4}\right)^{-}$ as $\frac{}{\left(\frac{}{4}\right)}$ using the negative exponent rule.
$g(-) = \frac{}{\left(\frac{}{4}\right)}$

STEP 10

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{}{\left(\frac{}{4}\right)}$ is equivalent to $\times4$ .
$g(-) = \times4$

STEP 11

Calculate the value of $\times4$ .
$g(-) =4$

STEP 12

Now, let's find $g(0)$ by substituting $0$ for $x$ in the function $g(x)$ .
$g(0) = \left(\frac{}{4}\right)^{0}$

STEP 13

Any number (except zero) raised to the power of zero is one.
$g(0) =$

STEP 14

Now, let's find $g()$ by substituting $$ for $x$ in the function $g(x)$.
$g() = \left(\frac{}{4}\right)^{}$

STEP 15

Any number raised to the power of one is the number itself.
$g() = \frac{}{4}$

STEP 16

Finally, let's find $g(2)$ by substituting $2$ for $x$ in the function $g(x)$ .
$g(2) = \left(\frac{}{4}\right)^{2}$

STEP 17

Calculate the value of $\left(\frac{}{4}\right)^2$ .
$g(2) = \frac{}{16}$ So, the completed table is\begin{tabular}{|c|c|} \hline $x$ & $g(x)$ \\ \hline-2 & $16$ \\ \hline- & $4$ \\ \hline0 & $$ \\ \hline & $\frac{}{4}$ \\ \hline2 & $\frac{}{16}$ \\ \hline\end{tabular}

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Find the values of $g(x) = (\frac{1}{4})^x$ for $x = -2, -1, 0, 1, 2$ .

STEP 1

Assumptions1. The function $g$ is defined as $g(x) = \left(\frac{1}{4}\right)^{x}$ . . We need to find the values of $g(x)$ for $x = -, -1,0,1,$ .

STEP 2

Let's start with $x = -2$ . We need to substitute $-2$ for $x$ in the function $g(x)$ .
$g(-2) = \left(\frac{1}{4}\right)^{-2}$

STEP 3

The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . So, we can rewrite $\left(\frac{1}{}\right)^{-2}$ as $\frac{1}{\left(\frac{1}{}\right)^2}$ .
$g(-2) = \frac{1}{\left(\frac{1}{}\right)^2}$

STEP 4

Calculate the value of $\left(\frac{1}{4}\right)^2$ .
$\left(\frac{1}{4}\right)^2 = \frac{1}{16}$

STEP 5

Substitute the value of $\left(\frac{1}{4}\right)^2$ back into the equation.
$g(-2) = \frac{1}{\frac{1}{16}}$

STEP 6

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{1}{\frac{1}{16}}$ is equivalent to $1 \times16$ .
$g(-2) =1 \times16$

STEP 7

Calculate the value of $1 \times16$ .
$g(-2) =16$

STEP 8

Now, let's find $g(-1)$ by substituting $-1$ for $x$ in the function $g(x)$ .
$g(-1) = \left(\frac{1}{4}\right)^{-1}$

STEP 9

Rewrite $\left(\frac{}{4}\right)^{-}$ as $\frac{}{\left(\frac{}{4}\right)}$ using the negative exponent rule.
$g(-) = \frac{}{\left(\frac{}{4}\right)}$

STEP 10

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{}{\left(\frac{}{4}\right)}$ is equivalent to $\times4$ .
$g(-) = \times4$

STEP 11

Calculate the value of $\times4$ .
$g(-) =4$

STEP 12

Now, let's find $g(0)$ by substituting $0$ for $x$ in the function $g(x)$ .
$g(0) = \left(\frac{}{4}\right)^{0}$

STEP 13

Any number (except zero) raised to the power of zero is one.
$g(0) =$

STEP 14

Now, let's find $g()$ by substituting $$ for $x$ in the function $g(x)$.
$g() = \left(\frac{}{4}\right)^{}$

STEP 15

Any number raised to the power of one is the number itself.
$g() = \frac{}{4}$

STEP 16

Finally, let's find $g(2)$ by substituting $2$ for $x$ in the function $g(x)$ .
$g(2) = \left(\frac{}{4}\right)^{2}$

STEP 17

Calculate the value of $\left(\frac{}{4}\right)^2$ .
$g(2) = \frac{}{16}$ So, the completed table is\begin{tabular}{|c|c|} \hline $x$ & $g(x)$ \\ \hline-2 & $16$ \\ \hline- & $4$ \\ \hline0 & $$ \\ \hline & $\frac{}{4}$ \\ \hline2 & $\frac{}{16}$ \\ \hline\end{tabular}

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Find the values of g(x)=(14)xg(x) = (\frac{1}{4})^xg(x)=(41​)x for x=−2,−1,0,1,2x = -2, -1, 0, 1, 2x=−2,−1,0,1,2.

STEP 1

Assumptions1. The function ggg is defined as g(x)=(14)xg(x) = \left(\frac{1}{4}\right)^{x}g(x)=(41​)x. . We need to find the values of g(x)g(x)g(x) for x=−,−1,0,1,x = -, -1,0,1,x=−,−1,0,1,.

STEP 2

Let's start with x=−2x = -2x=−2. We need to substitute −2-2−2 for xxx in the function g(x)g(x)g(x).g(−2)=(14)−2g(-2) = \left(\frac{1}{4}\right)^{-2}g(−2)=(41​)−2

STEP 3

The negative exponent rule states that a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​. So, we can rewrite (1)−2\left(\frac{1}{}\right)^{-2}(1​)−2 as 1(1)2\frac{1}{\left(\frac{1}{}\right)^2}(1​)21​.g(−2)=1(1)2g(-2) = \frac{1}{\left(\frac{1}{}\right)^2}g(−2)=(1​)21​

STEP 4

Calculate the value of (14)2\left(\frac{1}{4}\right)^2(41​)2.(14)2=116\left(\frac{1}{4}\right)^2 = \frac{1}{16}(41​)2=161​

STEP 5

Substitute the value of (14)2\left(\frac{1}{4}\right)^2(41​)2 back into the equation.g(−2)=1116g(-2) = \frac{1}{\frac{1}{16}}g(−2)=161​1​

STEP 6

The division of fractions is equivalent to multiplying by the reciprocal. So, 1116\frac{1}{\frac{1}{16}}161​1​ is equivalent to 1×161 \times161×16.g(−2)=1×16g(-2) =1 \times16g(−2)=1×16

STEP 7

Calculate the value of 1×161 \times161×16.g(−2)=16g(-2) =16g(−2)=16

STEP 8

Now, let's find g(−1)g(-1)g(−1) by substituting −1-1−1 for xxx in the function g(x)g(x)g(x).g(−1)=(14)−1g(-1) = \left(\frac{1}{4}\right)^{-1}g(−1)=(41​)−1

STEP 9

Rewrite (4)−\left(\frac{}{4}\right)^{-}(4​)− as (4)\frac{}{\left(\frac{}{4}\right)}(4​)​ using the negative exponent rule.g(−)=(4)g(-) = \frac{}{\left(\frac{}{4}\right)}g(−)=(4​)​

STEP 10

The division of fractions is equivalent to multiplying by the reciprocal. So, (4)\frac{}{\left(\frac{}{4}\right)}(4​)​ is equivalent to ×4 \times4×4.g(−)=×4g(-) = \times4g(−)=×4

STEP 11

Calculate the value of ×4 \times4×4.g(−)=4g(-) =4g(−)=4

STEP 12

Now, let's find g(0)g(0)g(0) by substituting 000 for xxx in the function g(x)g(x)g(x).g(0)=(4)0g(0) = \left(\frac{}{4}\right)^{0}g(0)=(4​)0

STEP 13

Any number (except zero) raised to the power of zero is one.g(0)=g(0) =g(0)=

STEP 14

Now, let's find g()g()g() by substituting $$ for $x$ in the function $g(x)$.g()=(4)g() = \left(\frac{}{4}\right)^{}g()=(4​)

STEP 15

Any number raised to the power of one is the number itself.g()=4g() = \frac{}{4}g()=4​

STEP 16

Finally, let's find g(2)g(2)g(2) by substituting 222 for xxx in the function g(x)g(x)g(x).g(2)=(4)2g(2) = \left(\frac{}{4}\right)^{2}g(2)=(4​)2

STEP 17

Calculate the value of (4)2\left(\frac{}{4}\right)^2(4​)2.g(2)=16g(2) = \frac{}{16}g(2)=16​So, the completed table is\begin{tabular}{|c|c|} \hlinexxx & g(x)g(x)g(x) \\ \hline-2 & 161616 \\ \hline- & 444 \\ \hline0 & $$ \\ \hline & $\frac{}{4}$ \\ \hline2 & $\frac{}{16}$ \\ \hline\end{tabular}

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Find the values of $g(x) = (\frac{1}{4})^x$ for $x = -2, -1, 0, 1, 2$ .

Assumptions1. The function $g$ is defined as $g(x) = \left(\frac{1}{4}\right)^{x}$ . . We need to find the values of $g(x)$ for $x = -, -1,0,1,$ .

Let's start with $x = -2$ . We need to substitute $-2$ for $x$ in the function $g(x)$ .
$g(-2) = \left(\frac{1}{4}\right)^{-2}$

The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$ . So, we can rewrite $\left(\frac{1}{}\right)^{-2}$ as $\frac{1}{\left(\frac{1}{}\right)^2}$ .
$g(-2) = \frac{1}{\left(\frac{1}{}\right)^2}$

Calculate the value of $\left(\frac{1}{4}\right)^2$ .
$\left(\frac{1}{4}\right)^2 = \frac{1}{16}$

Substitute the value of $\left(\frac{1}{4}\right)^2$ back into the equation.
$g(-2) = \frac{1}{\frac{1}{16}}$

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{1}{\frac{1}{16}}$ is equivalent to $1 \times16$ .
$g(-2) =1 \times16$

Calculate the value of $1 \times16$ .
$g(-2) =16$

Now, let's find $g(-1)$ by substituting $-1$ for $x$ in the function $g(x)$ .
$g(-1) = \left(\frac{1}{4}\right)^{-1}$

Rewrite $\left(\frac{}{4}\right)^{-}$ as $\frac{}{\left(\frac{}{4}\right)}$ using the negative exponent rule.
$g(-) = \frac{}{\left(\frac{}{4}\right)}$

The division of fractions is equivalent to multiplying by the reciprocal. So, $\frac{}{\left(\frac{}{4}\right)}$ is equivalent to $\times4$ .
$g(-) = \times4$

Calculate the value of $\times4$ .
$g(-) =4$

Now, let's find $g(0)$ by substituting $0$ for $x$ in the function $g(x)$ .
$g(0) = \left(\frac{}{4}\right)^{0}$

Any number (except zero) raised to the power of zero is one.
$g(0) =$

Now, let's find $g()$ by substituting $$ for $x$ in the function $g(x)$.
$g() = \left(\frac{}{4}\right)^{}$

Any number raised to the power of one is the number itself.
$g() = \frac{}{4}$

Finally, let's find $g(2)$ by substituting $2$ for $x$ in the function $g(x)$ .
$g(2) = \left(\frac{}{4}\right)^{2}$

Calculate the value of $\left(\frac{}{4}\right)^2$ .
$g(2) = \frac{}{16}$ So, the completed table is\begin{tabular}{|c|c|} \hline $x$ & $g(x)$ \\ \hline-2 & $16$ \\ \hline- & $4$ \\ \hline0 & $$ \\ \hline & $\frac{}{4}$ \\ \hline2 & $\frac{}{16}$ \\ \hline\end{tabular}