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

Nov 6, 2016

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

$4 = a {b}^{\frac{1}{2}}$
$2 \sqrt{2} = a {b}^{\frac{1}{4}}$

Solve for $a$ in equation $1$.

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

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

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

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

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

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

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

$b = 4$

Resubstitute to solve for $a$.

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

$4 = a \left(2\right)$

$a = 2$

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

Hopefully this helps!