Solved on Jan 11, 2024

Evaluate the function f(x)=3(13)x3f(x)=3\left(\frac{1}{3}\right)^{x}-3 at x=2x=-2. What is the numerical value of f(2)f(-2)?

STEP 1

Assumptions
1. The function is given as f(x)=3(13)x3 f(x) = 3\left(\frac{1}{3}\right)^{x} - 3
2. We need to evaluate the function at x=2 x = -2

STEP 2

Substitute x=2 x = -2 into the function f(x) f(x) .
f(2)=3(13)23 f(-2) = 3\left(\frac{1}{3}\right)^{-2} - 3

STEP 3

Recall that an=1an a^{-n} = \frac{1}{a^n} for any non-zero number a a and positive integer n n .

STEP 4

Apply the exponent rule from STEP_3 to the term (13)2 \left(\frac{1}{3}\right)^{-2} .
(13)2=1(13)2 \left(\frac{1}{3}\right)^{-2} = \frac{1}{\left(\frac{1}{3}\right)^2}

STEP 5

Calculate the square of 13 \frac{1}{3} .
(13)2=13×13=19 \left(\frac{1}{3}\right)^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}

STEP 6

Now, take the reciprocal of 19 \frac{1}{9} as indicated in STEP_4.
1(13)2=119=9 \frac{1}{\left(\frac{1}{3}\right)^2} = \frac{1}{\frac{1}{9}} = 9

STEP 7

Substitute the result from STEP_6 back into the function.
f(2)=3×93 f(-2) = 3 \times 9 - 3

STEP 8

Multiply 3 by 9.
3×9=27 3 \times 9 = 27

STEP 9

Subtract 3 from the result of STEP_8.
f(2)=273 f(-2) = 27 - 3

STEP 10

Calculate the final value of f(2) f(-2) .
f(2)=273=24 f(-2) = 27 - 3 = 24
The value of f(2) f(-2) is 24.

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