Solved on Jan 11, 2024

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

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Evaluate the function f(x)=3(13)x−3f(x)=3\left(\frac{1}{3}\right)^{x}-3f(x)=3(31​)x−3 at x=−2x=-2x=−2. What is the numerical value of f(−2)f(-2)f(−2)?

STEP 1

Assumptions1. The function is given as f(x)=3(13)x−3 f(x) = 3\left(\frac{1}{3}\right)^{x} - 3 f(x)=3(31​)x−32. We need to evaluate the function at x=−2 x = -2 x=−2

STEP 2

Substitute x=−2 x = -2 x=−2 into the function f(x) f(x) f(x).f(−2)=3(13)−2−3 f(-2) = 3\left(\frac{1}{3}\right)^{-2} - 3 f(−2)=3(31​)−2−3

STEP 3

Recall that a−n=1an a^{-n} = \frac{1}{a^n} a−n=an1​ for any non-zero number a a a and positive integer n n n.

STEP 4

Apply the exponent rule from STEP_3 to the term (13)−2 \left(\frac{1}{3}\right)^{-2} (31​)−2.(13)−2=1(13)2 \left(\frac{1}{3}\right)^{-2} = \frac{1}{\left(\frac{1}{3}\right)^2} (31​)−2=(31​)21​

STEP 5

Calculate the square of 13 \frac{1}{3} 31​.(13)2=13×13=19 \left(\frac{1}{3}\right)^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} (31​)2=31​×31​=91​

STEP 6

Now, take the reciprocal of 19 \frac{1}{9} 91​ as indicated in STEP_4.1(13)2=119=9 \frac{1}{\left(\frac{1}{3}\right)^2} = \frac{1}{\frac{1}{9}} = 9 (31​)21​=91​1​=9

STEP 7

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

STEP 8

Multiply 3 by 9.3×9=27 3 \times 9 = 27 3×9=27

STEP 9

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

STEP 10

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

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

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

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

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

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

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

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

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

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

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

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