Let $a$ and $b$ be positive real numbers and $b \neq 1$.
Find $a+2 b$ given that the exponential function $f(x)=a(b)^{x}$ passes through the points $P(0,3)$ and $Q\left(2, \frac{3}{16}\right)$.
a) $\frac{3}{8}$
b) $\frac{7}{2}$
c) $\frac{3}{16}$
d) $\frac{7}{4}$
e) 4
Answer
Finally, find $a + 2b = 3 + 2\left(\frac{1}{4}\right) = \boxed{\frac{5}{2}}$
Step 1 ：Given the points $P(0, 3)$ and $Q\left(2, \frac{3}{16}\right)$, we can create a system of equations using the function $f(x) = a(b)^x$:
Step 2 ：\[\begin{cases} a(b)^0 = 3 \\ a(b)^2 = \frac{3}{16} \end{cases}\]
Step 3 ：Solving the first equation, we get $a = 3$.
Step 4 ：Substitute $a = 3$ into the second equation: $3(b)^2 = \frac{3}{16}$.
Step 5 ：Solve for $b$: $b^2 = \frac{1}{16}$, so $b = \pm\frac{1}{4}$.
Step 6 ：Since $b \neq 1$, we choose $b = \frac{1}{4}$.
Step 7 ：Finally, find $a + 2b = 3 + 2\left(\frac{1}{4}\right) = \boxed{\frac{5}{2}}$
