Math / Algebra

Question1. Given the function $f(x)=\left(\frac{1}{4}\right)^{x}$ , evaluate each of the following. (a) $f(-3)$ (b) $f\left(-\frac{1}{2}\right)$ (c) $f(0)$

Studdy Solution

STEP 1

1. The function $f(x) = \left(\frac{1}{4}\right)^x$ is given.
2. To evaluate $f(x)$ for any value of $x$ , substitute the value of $x$ into the function and simplify.
3. The properties of exponents will be used to simplify the expressions.

STEP 2

1. Evaluate $f(-3)$ .
2. Evaluate $f\left(-\frac{1}{2}\right)$ .
3. Evaluate $f(0)$ .

STEP 3

Substitute $x = -3$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(-3) = \left(\frac{1}{4}\right)^{-3}$

STEP 4

Use the property of exponents that $a^{-b} = \frac{1}{a^b}$ to simplify the expression.
$f(-3) = \left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3 = 4^3$

STEP 5

Calculate $4^3$ .
$4^3 = 64$

STEP 6

Substitute $x = -\frac{1}{2}$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}}$

STEP 7

Use the property of exponents that $a^{-b} = \frac{1}{a^b}$ to simplify the expression.
$f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}} = \left(\frac{4}{1}\right)^{\frac{1}{2}} = 4^{\frac{1}{2}}$

STEP 8

Calculate $4^{\frac{1}{2}}$ , which is the square root of $4$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

STEP 9

Substitute $x = 0$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(0) = \left(\frac{1}{4}\right)^0$

STEP 10

Use the property of exponents that any non-zero number raised to the power of $0$ is $1$ .
$f(0) = 1$
Solution for the problem: (a) $f(-3) = 64$ (b) $f\left(-\frac{1}{2}\right) = 2$ (c) $f(0) = 1$

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Question1. Given the function $f(x)=\left(\frac{1}{4}\right)^{x}$ , evaluate each of the following. (a) $f(-3)$ (b) $f\left(-\frac{1}{2}\right)$ (c) $f(0)$

Studdy Solution

STEP 1

1. The function $f(x) = \left(\frac{1}{4}\right)^x$ is given.
2. To evaluate $f(x)$ for any value of $x$ , substitute the value of $x$ into the function and simplify.
3. The properties of exponents will be used to simplify the expressions.

STEP 2

1. Evaluate $f(-3)$ .
2. Evaluate $f\left(-\frac{1}{2}\right)$ .
3. Evaluate $f(0)$ .

STEP 3

Substitute $x = -3$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(-3) = \left(\frac{1}{4}\right)^{-3}$

STEP 4

Use the property of exponents that $a^{-b} = \frac{1}{a^b}$ to simplify the expression.
$f(-3) = \left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3 = 4^3$

STEP 5

Calculate $4^3$ .
$4^3 = 64$

STEP 6

Substitute $x = -\frac{1}{2}$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}}$

STEP 7

Use the property of exponents that $a^{-b} = \frac{1}{a^b}$ to simplify the expression.
$f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}} = \left(\frac{4}{1}\right)^{\frac{1}{2}} = 4^{\frac{1}{2}}$

STEP 8

Calculate $4^{\frac{1}{2}}$ , which is the square root of $4$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

STEP 9

Substitute $x = 0$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(0) = \left(\frac{1}{4}\right)^0$

STEP 10

Use the property of exponents that any non-zero number raised to the power of $0$ is $1$ .
$f(0) = 1$
Solution for the problem: (a) $f(-3) = 64$ (b) $f\left(-\frac{1}{2}\right) = 2$ (c) $f(0) = 1$

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Question1. Given the function f(x)=(14)xf(x)=\left(\frac{1}{4}\right)^{x}f(x)=(41​)x, evaluate each of the following. (a) f(−3)f(-3)f(−3) (b) f(−12)f\left(-\frac{1}{2}\right)f(−21​) (c) f(0)f(0)f(0)

Studdy Solution

STEP 1

1. The function f(x)=(14)xf(x) = \left(\frac{1}{4}\right)^xf(x)=(41​)x is given.2. To evaluate f(x)f(x)f(x) for any value of xxx, substitute the value of xxx into the function and simplify.3. The properties of exponents will be used to simplify the expressions.

STEP 2

1. Evaluate f(−3)f(-3)f(−3).2. Evaluate f(−12)f\left(-\frac{1}{2}\right)f(−21​).3. Evaluate f(0)f(0)f(0).

STEP 3

Substitute x=−3x = -3x=−3 into the function f(x)=(14)xf(x) = \left(\frac{1}{4}\right)^xf(x)=(41​)x.f(−3)=(14)−3 f(-3) = \left(\frac{1}{4}\right)^{-3} f(−3)=(41​)−3

STEP 4

Use the property of exponents that a−b=1aba^{-b} = \frac{1}{a^b}a−b=ab1​ to simplify the expression.f(−3)=(14)−3=(41)3=43 f(-3) = \left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3 = 4^3 f(−3)=(41​)−3=(14​)3=43

STEP 5

Calculate 434^343.43=64 4^3 = 64 43=64

STEP 6

Substitute x=−12x = -\frac{1}{2}x=−21​ into the function f(x)=(14)xf(x) = \left(\frac{1}{4}\right)^xf(x)=(41​)x.f(−12)=(14)−12 f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}} f(−21​)=(41​)−21​

STEP 7

STEP 8

Calculate 4124^{\frac{1}{2}}421​, which is the square root of 444.412=4=2 4^{\frac{1}{2}} = \sqrt{4} = 2 421​=4​=2

STEP 9

Substitute x=0x = 0x=0 into the function f(x)=(14)xf(x) = \left(\frac{1}{4}\right)^xf(x)=(41​)x.f(0)=(14)0 f(0) = \left(\frac{1}{4}\right)^0 f(0)=(41​)0

STEP 10

Use the property of exponents that any non-zero number raised to the power of 000 is 111.f(0)=1 f(0) = 1 f(0)=1Solution for the problem: (a) f(−3)=64f(-3) = 64f(−3)=64 (b) f(−12)=2f\left(-\frac{1}{2}\right) = 2f(−21​)=2 (c) f(0)=1f(0) = 1f(0)=1

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Question1. Given the function $f(x)=\left(\frac{1}{4}\right)^{x}$ , evaluate each of the following. (a) $f(-3)$ (b) $f\left(-\frac{1}{2}\right)$ (c) $f(0)$

1. The function $f(x) = \left(\frac{1}{4}\right)^x$ is given.
2. To evaluate $f(x)$ for any value of $x$ , substitute the value of $x$ into the function and simplify.
3. The properties of exponents will be used to simplify the expressions.

1. Evaluate $f(-3)$ .
2. Evaluate $f\left(-\frac{1}{2}\right)$ .
3. Evaluate $f(0)$ .

Substitute $x = -3$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(-3) = \left(\frac{1}{4}\right)^{-3}$

Use the property of exponents that $a^{-b} = \frac{1}{a^b}$ to simplify the expression.
$f(-3) = \left(\frac{1}{4}\right)^{-3} = \left(\frac{4}{1}\right)^3 = 4^3$

Calculate $4^3$ .
$4^3 = 64$

Substitute $x = -\frac{1}{2}$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f\left(-\frac{1}{2}\right) = \left(\frac{1}{4}\right)^{-\frac{1}{2}}$

Calculate $4^{\frac{1}{2}}$ , which is the square root of $4$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

Substitute $x = 0$ into the function $f(x) = \left(\frac{1}{4}\right)^x$ .
$f(0) = \left(\frac{1}{4}\right)^0$

Use the property of exponents that any non-zero number raised to the power of $0$ is $1$ .
$f(0) = 1$
Solution for the problem: (a) $f(-3) = 64$ (b) $f\left(-\frac{1}{2}\right) = 2$ (c) $f(0) = 1$