Answer :
Let's solve this step by step for each type of function using the given points [tex]\( f(4) = 6 \)[/tex] and [tex]\( f(8) = 18 \)[/tex].
a.) Linear Function: Assuming [tex]\( f(x) = mx + b \)[/tex]
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{f(8) - f(4)}{8 - 4} = \frac{18 - 6}{8 - 4} = \frac{12}{4} = 3
\][/tex]
2. Use the point-slope form to find [tex]\( b \)[/tex] using the point [tex]\( (4, 6) \)[/tex]:
[tex]\[
6 = 3 \times 4 + b \Rightarrow 6 = 12 + b \Rightarrow b = 6 - 12 = -6
\][/tex]
3. The function is [tex]\( f(x) = 3x - 6 \)[/tex]. Now calculate [tex]\( f(16) \)[/tex]:
[tex]\[
f(16) = 3 \times 16 - 6 = 48 - 6 = 42
\][/tex]
b.) Power Function: Assuming [tex]\( f(x) = ax^b \)[/tex]
1. Set up the equations:
[tex]\[
f(4) = a \times 4^b = 6 \quad \text{and} \quad f(8) = a \times 8^b = 18
\][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{8^b}{4^b} = \frac{18}{6} \Rightarrow 2^b = 3
\][/tex]
3. Solve for [tex]\( b \)[/tex] using [tex]\( 2^b = 3 \)[/tex]. Re-arrange to find [tex]\( b \approx \log_2(3) \)[/tex].
4. Use back substitution to find [tex]\( a \)[/tex] from [tex]\( a \times 4^b = 6 \)[/tex]:
[tex]\[
a = \frac{6}{4^b}
\][/tex]
5. Calculate [tex]\( f(16) = a \times 16^b \)[/tex]:
[tex]\[
f(16) = \frac{6}{4^b} \times 16^b = 6 \times \left(\frac{16}{4}\right)^b = 6 \times 4^{b} = 6 \times 3 = 54
\][/tex]
c.) Exponential Function: Assuming [tex]\( f(x) = a \times b^x \)[/tex]
1. Use the two points to set up equations:
[tex]\[
\frac{f(8)}{f(4)} = \frac{a \times b^8}{a \times b^4} = b^4 = \frac{18}{6} = 3
\][/tex]
2. Solve for [tex]\( b \)[/tex] by taking the fourth root:
[tex]\[
b = \sqrt[4]{3}
\][/tex]
3. Use one point to solve for [tex]\( a \)[/tex]:
[tex]\[
f(4) = a \times b^4 = 6 \quad \Rightarrow a = \frac{6}{b^4} = 2
\][/tex]
4. Calculate [tex]\( f(16) \)[/tex]:
[tex]\[
f(16) = a \times b^{16} = 2 \times (b^4)^4 = 2 \times 3^4 = 2 \times 81 = 162
\][/tex]
We have solved parts (a) through (c) with [tex]\( f(16) \)[/tex] being 42 for linear, 54 for power, and 162 for exponential. For the logarithmic case (d), please consult additional resources or information, as it requires specific methods beyond the information provided.
a.) Linear Function: Assuming [tex]\( f(x) = mx + b \)[/tex]
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[
m = \frac{f(8) - f(4)}{8 - 4} = \frac{18 - 6}{8 - 4} = \frac{12}{4} = 3
\][/tex]
2. Use the point-slope form to find [tex]\( b \)[/tex] using the point [tex]\( (4, 6) \)[/tex]:
[tex]\[
6 = 3 \times 4 + b \Rightarrow 6 = 12 + b \Rightarrow b = 6 - 12 = -6
\][/tex]
3. The function is [tex]\( f(x) = 3x - 6 \)[/tex]. Now calculate [tex]\( f(16) \)[/tex]:
[tex]\[
f(16) = 3 \times 16 - 6 = 48 - 6 = 42
\][/tex]
b.) Power Function: Assuming [tex]\( f(x) = ax^b \)[/tex]
1. Set up the equations:
[tex]\[
f(4) = a \times 4^b = 6 \quad \text{and} \quad f(8) = a \times 8^b = 18
\][/tex]
2. Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{8^b}{4^b} = \frac{18}{6} \Rightarrow 2^b = 3
\][/tex]
3. Solve for [tex]\( b \)[/tex] using [tex]\( 2^b = 3 \)[/tex]. Re-arrange to find [tex]\( b \approx \log_2(3) \)[/tex].
4. Use back substitution to find [tex]\( a \)[/tex] from [tex]\( a \times 4^b = 6 \)[/tex]:
[tex]\[
a = \frac{6}{4^b}
\][/tex]
5. Calculate [tex]\( f(16) = a \times 16^b \)[/tex]:
[tex]\[
f(16) = \frac{6}{4^b} \times 16^b = 6 \times \left(\frac{16}{4}\right)^b = 6 \times 4^{b} = 6 \times 3 = 54
\][/tex]
c.) Exponential Function: Assuming [tex]\( f(x) = a \times b^x \)[/tex]
1. Use the two points to set up equations:
[tex]\[
\frac{f(8)}{f(4)} = \frac{a \times b^8}{a \times b^4} = b^4 = \frac{18}{6} = 3
\][/tex]
2. Solve for [tex]\( b \)[/tex] by taking the fourth root:
[tex]\[
b = \sqrt[4]{3}
\][/tex]
3. Use one point to solve for [tex]\( a \)[/tex]:
[tex]\[
f(4) = a \times b^4 = 6 \quad \Rightarrow a = \frac{6}{b^4} = 2
\][/tex]
4. Calculate [tex]\( f(16) \)[/tex]:
[tex]\[
f(16) = a \times b^{16} = 2 \times (b^4)^4 = 2 \times 3^4 = 2 \times 81 = 162
\][/tex]
We have solved parts (a) through (c) with [tex]\( f(16) \)[/tex] being 42 for linear, 54 for power, and 162 for exponential. For the logarithmic case (d), please consult additional resources or information, as it requires specific methods beyond the information provided.
