Write the function that represents the sequence in the table.

Question

Start 2 By 5 Table 1st Row 1st Column x 2nd Column 1 3rd Column 2 4st Column 3 5st Column 4 2nd Row 1st Column f left parenthesis x right parenthesis 2nd Column 6 3rd Column 24 4st Column 96 5st Column 384 EndTable

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Part 1
The function that represents the sequence in the table is f left parenthesis x right parenthesis equals
  
enter your response here.
​(Type your answer using exponential​ notation.)

Answer 1

To represent the sequence in the table you've provided, we can analyze the values in the second row:

• For $x=1$, $f(1)=6$
• For $x=2$, $f(2)=24$
• For $x=3$, $f(3)=96$
• For $x=4$, $f(4)=384$

Let's examine the pattern in the function values:

1. Find the ratio of consecutive values:
   • $\frac{f(2)}{f(1)}=\frac{24}{6}=4$
   • $\frac{f(3)}{f(2)}=\frac{96}{24}=4$
   • $\frac{f(4)}{f(3)}=\frac{384}{96}=4$

The function values are multiplied by 4 as $x$ increases by 1, suggesting an exponential relationship.

2. Check if the function fits a standard form: Given the common ratio, we can hypothesize that $f(x)=k\cdot 4^{x-1}$ for some constant $k$.

3. Determine $k$:

   • We can find $k$ using the value of $f(1)$:
     • $f(1)=k\cdot 4^{1-1}=k\cdot 1=k$
     • Since $f(1)=6$, we find that $k=6$.

Thus, we have the function:

$$f(x)=6\cdot 4^{x-1}$$

This matches the values given in the table. Therefore, the function that represents the sequence in the table is:

$$\boxed{{\displaystyle 6\cdot 4^{x-1}}}$$