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The functions f(x)=8((2)/(5))^(x) and g(x)=8(b)^(x) are graphed in the xy-plane. For what value of
b would the graphs of functions f and g be symmetric with respect to the y-axis?

The functions $f(x)=8\left(\frac{2}{5}\right)^{x}$ and $g(x)=8(b)^{x}$ are graphed in the $y$ -plane. For what value of $b$ would the graphs of functions $f$ and $g$ be symmetric with respect to the $y$ -axis?

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Evaluate expression when two complex numbers are given

Full solution

Q. The functions

f(x)=8\left(\frac{2}{5}\right)^{x}

and

g(x)=8(b)^{x}

are graphed in the

y

-plane. For what value of

b

would the graphs of functions

f

and

g

be symmetric with respect to the

y

-axis?

Set Equations Equal: To be symmetric with respect to the y-axis, the functions must be reflections of each other across the y-axis. This means that $f(x)$ must equal $g(-x)$ for all $x$ .
Solve for $b$ : Let's set $f(x)$ equal to $g(-x)$ and solve for $b$ . $8\left(\frac{2}{5}\right)^x = 8b^{-x}$
Simplify Equation: Divide both sides by $8$ to simplify the equation. $(\frac{2}{5})^x = b^{-x}$
Recognize Base Relationship: Recognize that $b^{-x}$ is the same as $(1/b)^x$ . $(2/5)^x = (1/b)^x$
Find Value of $b$ : Since the bases are being raised to the same power, the bases themselves must be equal for the equation to hold true for all $x$ . $\frac{2}{5} = \frac{1}{b}$
Find Value of b: Since the bases are being raised to the same power, the bases themselves must be equal for the equation to hold true for all $x$ . $\frac{2}{5} = \frac{1}{b}$ Solve for $b$ by taking the reciprocal of both sides. $b = \frac{5}{2}$

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