Chapter 1.3, Problem 48E

To determine

To find:The value of $a$ and $k$ such that the exponential function $f\left(x\right)=k\cdot a^x$ passes through the points $\left(1,4.5\right)$ and $\left(-1,0.5\right)$ .

Answer to Problem 48E

The value of $a$ is $3$ and $k$ is $1.5$ .

Explanation of Solution

Given information:The function $f\left(x\right)=k\cdot a^x$ passes through the points $\left(1,4.5\right)$ and $\left(-1,0.5\right)$ .

Calculation:

As the function $f\left(x\right)=k\cdot a^x$ passes through the points $\left(1,4.5\right)$ and $\left(-1,0.5\right)$ .

Substitute $1$ for $x$ and $4.5$ for $f\left(x\right)$ in the given function.

  $\begin{array}{c}4.5=k\cdot a^1\\ 4.5=k\cdot a\end{array}$ ................(1)

Substitute $-1$ for $x$ and $0.5$ for $f\left(x\right)$ in the given function.

  $0.5=k\cdot a^{-1}$ ................(2)

Divide equation $\left(1\right)$ by the equation $\left(2\right)$ .

  $\begin{array}{c}\frac{4.5}{0.5}=\frac{k\cdot a}{k\cdot a^{-1}}\\ 9=a^{1+1}\\ a^2=9\\ a=\pm 3\end{array}$

For the exponential function take only $a>0$ . So, $a=-3$ is neglected.

Substitute $3$ for $a$ in equation $\left(1\right)$ .

  $\begin{array}{c}4.5=k\cdot 3\\ k=\frac{4.5}{3}\\ k=1.5\end{array}$

Therefore, the value of $a$ is $3$ and $k$ is $1.5$ .