Question
This table shows some values of the function g(x):
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.
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.
Answers
GPT-4o mini
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:
\[
g(x) = \frac{3}{2} \cdot 2^x
\]
\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-1 & 3 \\
0 & 2 \\
1 & 3 \\
2 & 6 \\
3 & 12 \\
\hline
\end{array}
\]
### 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:
\[
g(x) = \frac{3}{2} \cdot 2^x
\]