Answer :
To complete the table for [tex]\( h(x) \)[/tex] and find the equation for [tex]\( h(x) \)[/tex], let's follow a structured approach.
### Step-by-Step Solution:
1. Identify Given Values:
- The values of [tex]\( h(x) \)[/tex] for [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex] are provided:
[tex]\[ h(1) = 10, \][/tex]
[tex]\[ h(2) = 40. \][/tex]
2. Determine the form of the function:
- We hypothesize that [tex]\( h(x) \)[/tex] is an exponential function of the form:
[tex]\[ h(x) = a \cdot b^x. \][/tex]
3. Find constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- Using the values [tex]\( h(1) = 10 \)[/tex] and [tex]\( h(2) = 40 \)[/tex]:
[tex]\[ h(1) = a \cdot b^1 = 10 \implies a \cdot b = 10, \][/tex]
[tex]\[ h(2) = a \cdot b^2 = 40 \implies a \cdot b^2 = 40. \][/tex]
- Solve these equations to find [tex]\( b \)[/tex]:
- From [tex]\( a \cdot b = 10 \)[/tex]:
[tex]\[ a = \frac{10}{b}. \][/tex]
- Substitute [tex]\( a \)[/tex] in the second equation:
[tex]\[ \left(\frac{10}{b}\right) \cdot b^2 = 40 \implies 10b = 40 \implies b = 4. \][/tex]
- Using [tex]\( b = 4 \)[/tex], find [tex]\( a \)[/tex]:
[tex]\[ a \cdot 4 = 10 \implies a = 2.5. \][/tex]
4. Construct the exponential function:
- Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], the equation for [tex]\( h(x) \)[/tex] is:
[tex]\[ h(x) = 10 \cdot 4^x. \][/tex]
5. Complete the table:
- Using [tex]\( h(x) = 10 \cdot 4^x \)[/tex], calculate the missing values for [tex]\( x = 0, 3, 4 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = 10 \cdot 4^0 = 10 \cdot 1 = 10. \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = 10 \cdot 4^3 = 10 \cdot 64 = 640. \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = 10 \cdot 4^4 = 10 \cdot 256 = 2560. \][/tex]
6. Filled Table:
[tex]\[ \begin{tabular}{|l|l|} \hline $x$ & $h(x)$ \\ \hline 0 & 10 \\ \hline 1 & 10 \\ \hline 2 & 40 \\ \hline 3 & 640 \\ \hline 4 & 2560 \\ \hline \end{tabular} \][/tex]
7. Equation for [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = 10 \cdot 4^x. \][/tex]
### Step-by-Step Solution:
1. Identify Given Values:
- The values of [tex]\( h(x) \)[/tex] for [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex] are provided:
[tex]\[ h(1) = 10, \][/tex]
[tex]\[ h(2) = 40. \][/tex]
2. Determine the form of the function:
- We hypothesize that [tex]\( h(x) \)[/tex] is an exponential function of the form:
[tex]\[ h(x) = a \cdot b^x. \][/tex]
3. Find constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- Using the values [tex]\( h(1) = 10 \)[/tex] and [tex]\( h(2) = 40 \)[/tex]:
[tex]\[ h(1) = a \cdot b^1 = 10 \implies a \cdot b = 10, \][/tex]
[tex]\[ h(2) = a \cdot b^2 = 40 \implies a \cdot b^2 = 40. \][/tex]
- Solve these equations to find [tex]\( b \)[/tex]:
- From [tex]\( a \cdot b = 10 \)[/tex]:
[tex]\[ a = \frac{10}{b}. \][/tex]
- Substitute [tex]\( a \)[/tex] in the second equation:
[tex]\[ \left(\frac{10}{b}\right) \cdot b^2 = 40 \implies 10b = 40 \implies b = 4. \][/tex]
- Using [tex]\( b = 4 \)[/tex], find [tex]\( a \)[/tex]:
[tex]\[ a \cdot 4 = 10 \implies a = 2.5. \][/tex]
4. Construct the exponential function:
- Now that we have [tex]\( a \)[/tex] and [tex]\( b \)[/tex], the equation for [tex]\( h(x) \)[/tex] is:
[tex]\[ h(x) = 10 \cdot 4^x. \][/tex]
5. Complete the table:
- Using [tex]\( h(x) = 10 \cdot 4^x \)[/tex], calculate the missing values for [tex]\( x = 0, 3, 4 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = 10 \cdot 4^0 = 10 \cdot 1 = 10. \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = 10 \cdot 4^3 = 10 \cdot 64 = 640. \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = 10 \cdot 4^4 = 10 \cdot 256 = 2560. \][/tex]
6. Filled Table:
[tex]\[ \begin{tabular}{|l|l|} \hline $x$ & $h(x)$ \\ \hline 0 & 10 \\ \hline 1 & 10 \\ \hline 2 & 40 \\ \hline 3 & 640 \\ \hline 4 & 2560 \\ \hline \end{tabular} \][/tex]
7. Equation for [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = 10 \cdot 4^x. \][/tex]
