Answer :
Let's tackle this step by step to find the [tex]\( y \)[/tex]-intercept of the exponential function [tex]\( f(x) \)[/tex].
Given the data points:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 12 & 24 & 48 & 96 \\ \hline \end{array} \][/tex]
First, note that the function is an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex].
### Step 1: Determine the ratio between successive [tex]\( f(x) \)[/tex] values
Calculate the ratio [tex]\( b \)[/tex] using consecutive values:
[tex]\[ \frac{f(3)}{f(2)} = \frac{24}{12} = 2 \][/tex]
[tex]\[ \frac{f(4)}{f(3)} = \frac{48}{24} = 2 \][/tex]
[tex]\[ \frac{f(5)}{f(4)} = \frac{96}{48} = 2 \][/tex]
Since the ratio [tex]\( b \)[/tex] is constant, the exponential function has a base [tex]\( b = 2 \)[/tex].
### Step 2: Determine the coefficient [tex]\( a \)[/tex]
Using one of the points, say [tex]\( x = 2 \)[/tex] and [tex]\( f(2) = 12 \)[/tex], we can find [tex]\( a \)[/tex]:
[tex]\[ f(2) = a \cdot b^2 \implies 12 = a \cdot 2^2 \implies 12 = a \cdot 4 \implies a = \frac{12}{4} = 3 \][/tex]
Thus, the function is:
[tex]\[ f(x) = 3 \cdot 2^x \][/tex]
### Step 3: Identify the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3 \][/tex]
### Step 4: Graph [tex]\( f(x) \)[/tex]
We can plot the points from the table and verify that they fit on the graph of [tex]\( f(x) = 3 \cdot 2^x \)[/tex]. The key characteristics include the exponential growth and the [tex]\( y \)[/tex]-intercept at 3.
### Conclusion
The [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
Given the data points:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 2 & 3 & 4 & 5 \\ \hline f(x) & 12 & 24 & 48 & 96 \\ \hline \end{array} \][/tex]
First, note that the function is an exponential function of the form [tex]\( f(x) = a \cdot b^x \)[/tex].
### Step 1: Determine the ratio between successive [tex]\( f(x) \)[/tex] values
Calculate the ratio [tex]\( b \)[/tex] using consecutive values:
[tex]\[ \frac{f(3)}{f(2)} = \frac{24}{12} = 2 \][/tex]
[tex]\[ \frac{f(4)}{f(3)} = \frac{48}{24} = 2 \][/tex]
[tex]\[ \frac{f(5)}{f(4)} = \frac{96}{48} = 2 \][/tex]
Since the ratio [tex]\( b \)[/tex] is constant, the exponential function has a base [tex]\( b = 2 \)[/tex].
### Step 2: Determine the coefficient [tex]\( a \)[/tex]
Using one of the points, say [tex]\( x = 2 \)[/tex] and [tex]\( f(2) = 12 \)[/tex], we can find [tex]\( a \)[/tex]:
[tex]\[ f(2) = a \cdot b^2 \implies 12 = a \cdot 2^2 \implies 12 = a \cdot 4 \implies a = \frac{12}{4} = 3 \][/tex]
Thus, the function is:
[tex]\[ f(x) = 3 \cdot 2^x \][/tex]
### Step 3: Identify the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3 \][/tex]
### Step 4: Graph [tex]\( f(x) \)[/tex]
We can plot the points from the table and verify that they fit on the graph of [tex]\( f(x) = 3 \cdot 2^x \)[/tex]. The key characteristics include the exponential growth and the [tex]\( y \)[/tex]-intercept at 3.
### Conclusion
The [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
