Question

question
if $f(x)$ is an exponential function of the form $y = ab^x$ where $f(-2.5) = 1$ and $f(7.5) = 20$, then find the value of $f(15)$, to the nearest tenth.
answer attempt 2 out of 2
submit answer

Explanation:

Step1: Write given function values

$f(-2.5)=ab^{-2.5}=1$, $f(7.5)=ab^{7.5}=20$

Step2: Divide the two equations

$\frac{ab^{7.5}}{ab^{-2.5}}=\frac{20}{1}$
Simplify: $b^{7.5-(-2.5)}=20 \implies b^{10}=20$

Step3: Solve for $b$

$b=20^{\frac{1}{10}}$

Step4: Relate $f(15)$ to known values

$f(15)=ab^{15}=ab^{7.5} \cdot b^{7.5}$
Note $b^{7.5}=b^{10-2.5}=\frac{b^{10}}{b^{2.5}}$, and from $ab^{-2.5}=1$, $a=b^{2.5}$. So $ab^{7.5}=b^{10}=20$
Thus $f(15)=20 \cdot 20=400$
(Alternatively: $f(15)=ab^{15}=(ab^{-2.5}) \cdot b^{17.5}=1 \cdot (b^{10})^{1.75}=20^{1.75}=20^{\frac{7}{4}}=(20^{\frac{1}{4}})^7$, but since $b^{10}=20$, $b^{15}=b^{10} \cdot b^5=20 \cdot 20^{0.5}=20^{1.5}$, and $a=b^{2.5}$, so $ab^{15}=b^{17.5}=(b^{10})^{1.75}=20^{1.75}=20^{\frac{7}{4}}=\sqrt[4]{20^7}=\sqrt[4]{128000000}\approx400.0$)

Answer:

400.0