Chapter 1.3, Problem 49E

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 $\left(1,1.5\right)$ and $\left(-1,6\right)$ .

Answer to Problem 49E

The value of $a$ is $0.5$ and $k$ is $3$ .

Explanation of Solution

Given information:The function $f\left(x\right)=k\cdot a^x$ passes through the points $\left(1,1.5\right)$ and $\left(-1,6\right)$ .

Calculation:

As the function $f\left(x\right)=k\cdot a^x$ passes through the points $\left(1,1.5\right)$ and $\left(-1,6\right)$ .

Substitute $1$ for $x$ and $1.5$ for $f\left(x\right)$ in the given function.

  $\begin{array}{c}1.5=k\cdot a^1\\ 1.5=k\cdot a\end{array}$................(1)

Substitute $-1$ for $x$ and $6$ for $f\left(x\right)$ in the given function.

  $6=k\cdot a^{-1}$................(2)

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

  $\begin{array}{c}\frac{1.5}{6}=\frac{k\cdot a}{k\cdot a^{-1}}\\ 0.25=a^{1+1}\\ a^2=0.25\\ a=\pm 0.5\end{array}$

For the exponential function take only $a>0$ . So, $a=-0.5$ is neglected.

Substitute $0.5$ for $a$ in equation $\left(1\right)$ .

  $\begin{array}{c}1.5=k\cdot \left(0.5\right)\\ k=\frac{1.5}{0.5}\\ k=3\end{array}$

Therefore, the value of $a$ is $0.5$ and $k$ is $3$ .