Question

Please help with homework!Answer Options are: f(x), k(x), h(x), g(x)

Answer 1

f(x), h(x) and k(x) can not be an exponential function.

Step-by-step explanation:

A exponential function can be represented by:

`f(x)=a\cdot b^x`

Let's evaluate each function and see which of the functions fit the expression above.

First, let's evaluate x = 0

For x = 0

f(x) = 3

g(x) = 1

h(x) = 1

k(x) = 0

If we substitute x=0 in the expression above, we will find that:

`\begin{gathered} f(x)=a\cdot b^x \\ 3=a\cdot b^0 \\ 3=a \\ \\ g(x)=a\cdot b^x \\ 1=a\cdot b^0 \\ 1=a \\ \\ h(x) \\ 1=a \\ \\ k(x) \\ 0=a\cdot b^0 \\ 0=a \end{gathered}`

To be consider a exponential function, a can not be zero. Thus, k(x) can not be an exponential function

Now, we already now the value for "a". The next step is to find the value for b:

Let's evaluate x = 1

`\begin{gathered} f(x)=a\cdot b^x \\ f(x)=3\cdot b^x \\ 4.95=3\cdot b^1 \\ b=(4.95)/(3) \\ b=1.65 \\ \\ g(x)=1\cdot b^x \\ 2=b^1 \\ b=2 \\ \\ h(x)=1\cdot b^x \\ 1.25=b^1 \\ b=1.25 \end{gathered}`

Now, we can write all the posible exponential functions. Then, we can test the other given points:

`\begin{gathered} f(x)=1.65\cdot3^x \\ For\text{ x=2} \\ f(x)=1.65\cdot3^2 \\ f(x)=1.65\cdot9 \\ f(x)=14.85 \end{gathered}`

As we can see, f(x) in not an exponential function.

`\begin{gathered} g(x)=2^x \\ \text{for x=2},\text{ g(x)=}2^2=4 \\ \text{for x=3},\text{ g(x)=}2^3=8 \\ \text{for x=-1, g(x)=}2^(-1)=0.5 \end{gathered}`

All the points are the same as the presented in the table.

As we can see, g(x) can be an exponential function.

`\begin{gathered} h(x)=1.25^x \\ \text{for x=2, h(x)=}1.25^2=1.56 \\ \text{for x=3, h(x)=1.25}^3=1.95 \end{gathered}`

As we can see, g(x) can not be an exponential function.