How do you find formulas for the exponential functions satisfying the given conditions g(1/2)=4 and g(1/4)=2sqrt2?

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How do you find formulas for the exponential functions satisfying the given conditions g(1/2)=4 and g(1/4)=2sqrt2?

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Answer 1 · 2016-11-06T22:52:45

An exponential function is generally of the form $y = a b^{x}$ $y = ab^x$ . So, knowing the inputs/outputs of the function, we can write a system of equations with respect to $a$ $a$ and $b$ $b$ .

$4 = a b^{\frac{1}{2}}$ $4 = ab^(1/2)$
$2 \sqrt{2} = a b^{\frac{1}{4}}$ $2sqrt(2) = ab^(1/4)$

Solve for $a$ $a$ in equation $1$ $1$ .

$a = \frac{4}{b^{\frac{1}{2}}}$ $a = 4/(b^(1/2))$

$2 \sqrt{2} = \frac{4}{b^{\frac{1}{2}}} b^{\frac{1}{4}}$ $2sqrt(2) = 4/(b^(1/2))b^(1/4)$

$2 \sqrt{2} = \frac{4}{b^{\frac{1}{4}}}$ $2sqrt(2) = 4/b^(1/4)$

$b^{\frac{1}{4}} = \frac{4}{2 \sqrt{2}}$ $b^(1/4) = 4/(2sqrt(2))$

$b^{\frac{1}{4}} = \frac{2}{\sqrt{2}}$ $b^(1/4) = 2/sqrt(2)$

$b = {(\frac{2}{\sqrt{2}})}^{4}$ $b = (2/sqrt(2))^4$

$b = \frac{16}{4}$ $b= 16/4$

$b = 4$ $b = 4$

Resubstitute to solve for $a$ $a$ .

$4 = a {(4)}^{\frac{1}{2}}$ $4 = a(4)^(1/2)$

$4 = a (2)$ $4 = a(2)$

$a = 2$ $a = 2$

Hence, the equation is $y = 2 {(4)}^{x}$ $y = 2(4)^x$ .

Hopefully this helps!

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How do you find formulas for the exponential functions satisfying the given conditions g(1/2)=4 and g(1/4)=2sqrt2?

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