Solved on Nov 27, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

The division of fractions is equivalent to multiplying by the reciprocal. So, 1116\frac{1}{\frac{1}{16}} is equivalent to 1×161 \times16.
g(2)=1×16g(-2) =1 \times16

STEP 7

Calculate the value of 1×161 \times16.
g(2)=16g(-2) =16

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

Now, let's find g(0)g(0) by substituting 00 for xx in the function g(x)g(x).
g(0)=(4)0g(0) = \left(\frac{}{4}\right)^{0}

STEP 13

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

STEP 14

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

STEP 15

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

STEP 16

Finally, let's find g(2)g(2) by substituting 22 for xx in the function g(x)g(x).
g(2)=(4)2g(2) = \left(\frac{}{4}\right)^{2}

STEP 17

Calculate the value of (4)2\left(\frac{}{4}\right)^2.
g(2)=16g(2) = \frac{}{16}So, the completed table is\begin{tabular}{|c|c|} \hlinexx & g(x)g(x) \\ \hline-2 & 1616 \\ \hline- & 44 \\ \hline0 & $$ \\ \hline & $\frac{}{4}$ \\ \hline2 & $\frac{}{16}$ \\ \hline\end{tabular}

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord