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 \( g(x) \) is linear or exponential, we can analyze the values provided in the table:

\[ \begin{array}{|c|c|} \hline x & g(x) \ \hline -1 & 3 \ 0 & 2 \ 1 & 3 \ 2 & 6 \ 3 & 12 \ \hline \end{array} \]

Step 1: Check if \( g(x) \) is linear

For a function to be linear, the differences between consecutive \( g(x) \) values should be constant. Let's calculate these differences:

• \( g(0) - g(-1) = 2 - 3 = -1 \)
• \( g(1) - g(0) = 3 - 2 = 1 \)
• \( g(2) - g(1) = 6 - 3 = 3 \)
• \( g(3) - g(2) = 12 - 6 = 6 \)

The first differences are \(-1, 1, 3, 6\), which are not constant. Thus, \( g(x) \) is not linear.

Step 2: Check if \( g(x) \) is exponential

For a function to be exponential, the ratios of consecutive \( g(x) \) values should be constant. Let's calculate these ratios:

• \( \frac{g(0)}{g(-1)} = \frac{2}{3} \)
• \( \frac{g(1)}{g(0)} = \frac{3}{2} \)
• \( \frac{g(2)}{g(1)} = \frac{6}{3} = 2 \)
• \( \frac{g(3)}{g(2)} = \frac{12}{6} = 2 \)

The ratios are \(\frac{2}{3}, \frac{3}{2}, 2, 2\). The ratios are not constant, but let’s see if we can identify a consistent pattern in the values of \( g(x) \):

Step 3: Examining possible exponential growth

Looking more closely at the values could reveal an exponential pattern:

• Notice that starting from \( g(1) = 3 \), \( g(2) = 6 \) (which is \( 2 \times 3 \)) and \( g(3) = 12 \) (which is \( 2 \times 6 \)).

Step 4: Hypothesis of the form

This suggests a potential form for the function:

• It appears that:
  • From \( g(1) \) to \( g(2) \), we multiply by 2.
  • From \( g(2) \) to \( g(3) \), we also multiply by 2.

To model this, we can set \( g(x) = a \cdot b^x \).

Step 5: Finding parameters \( a \) and \( b \)

From the table values:

• Let’s take the point \( g(1) = 3 \):

\[ 3 = a \cdot b^1 \rightarrow a \cdot b = 3 \quad (1) \]

• Now, for \( g(2) = 6 \):

\[ 6 = a \cdot b^2 \quad (2) \]

Dividing (2) by (1): \[ \frac{6}{3} = \frac{a \cdot b^2}{a \cdot b} \quad \Rightarrow 2 = b \]

Putting \( b = 2 \) back into (1): \[ a \cdot 2 = 3 \quad \Rightarrow a = \frac{3}{2} \]

Conclusion: Write the equation

Thus, the function \( g(x) \) can be expressed as:

\[ g(x) = \frac{3}{2} \cdot 2^x \]

This confirms that \( g(x) \) is an exponential function. Therefore, the final answer is:

\[ g(x) = \frac{3}{2} \cdot 2^x \]