This table shows some values of the function g(x):

Question

x –1 0 1 2 3
g(x) 3/2 3 6 12 24
Is g(x) linear or exponential?
Write an equation for g(x). If it is linear, write it in the form g(x)=mx+b. If it is exponential, write it in the form g(x)=a(b)x.

Answer 1

To determine whether the function $g(x)$ is linear or exponential, we can analyze the growth of the values given for $g(x)$.

Step 1: Check for Linearity

A linear function has a constant rate of change (slope). We can calculate the differences between consecutive values of $g(x)$:

$$\begin{array}{rlrlr}g(0)-g(-1)& =3-\frac{3}{2}=\frac{3}{2}=1.5g(1)-g(0)& =6-3=3g(2)-g(1)& =12-6=6g(3)-g(2)& =24-12=12\end{array}$$

The differences are:

• $1.5$ (from -1 to 0)
• $3$ (from 0 to 1)
• $6$ (from 1 to 2)
• $12$ (from 2 to 3)

Since the differences (1.5, 3, 6, 12) are not constant, $g(x)$ is not linear.

Step 2: Check for Exponential Growth

An exponential function has a constant ratio between consecutive values. We can calculate the ratios of consecutive values of $g(x)$:

$$\begin{array}{rlrlr}\frac{g(0)}{g(-1)}& =\frac{3}{\frac{3}{2}}=2\frac{g(1)}{g(0)}& =\frac{6}{3}=2\frac{g(2)}{g(1)}& =\frac{12}{6}=2\frac{g(3)}{g(2)}& =\frac{24}{12}=2\end{array}$$

The ratios are all equal to $2$, which indicates that $g(x)$ exhibits exponential growth.

Step 3: Write the Exponential Equation

Since $g(x)$ can be expressed in exponential form, we can use the following general equation:

$$g(x)=a(b^x)$$

Given the ratios, we can identify $b=2$. Let's find $a$ using one of the known points. We can use $g(0)=3$:

$$g(0)=a(2^0)=a(1)=a⟹a=3$$

Thus, we can write the function as:

$$g(x)=3(2^x)$$

Final Result

The function $g(x)$ is exponential, and the equation in the desired form is:

$$g(x)=3(2^x)$$