Answer :
To solve the problem, we need to find the value of [tex]\( f(2.5) \)[/tex] given that [tex]\( f(x) = a \cdot b^x \)[/tex] with two conditions: [tex]\( f(-1) = 2 \)[/tex] and [tex]\( f(2) = 84 \)[/tex].
Step 1: Use the given conditions to create equations
1. [tex]\( f(-1) = 2 \)[/tex] means [tex]\( a \cdot b^{-1} = 2 \)[/tex]. This equation can be rewritten as:
[tex]\[
\frac{a}{b} = 2 \quad \Rightarrow \quad a = 2b
\][/tex]
2. [tex]\( f(2) = 84 \)[/tex] means [tex]\( a \cdot b^2 = 84 \)[/tex].
Step 2: Substitute and solve for [tex]\( b \)[/tex]
Substitute [tex]\( a = 2b \)[/tex] from the first equation into the second equation:
[tex]\[
(2b) \cdot b^2 = 84 \quad \Rightarrow \quad 2b^3 = 84
\][/tex]
Divide both sides by 2:
[tex]\[
b^3 = 42
\][/tex]
Find [tex]\( b \)[/tex] by taking the cube root of 42:
[tex]\[
b = \sqrt[3]{42}
\][/tex]
Step 3: Solve for [tex]\( a \)[/tex] using the value of [tex]\( b \)[/tex]
Since [tex]\( a = 2b \)[/tex], we can substitute the value of [tex]\( b \)[/tex]:
[tex]\[
a = 2 \cdot \sqrt[3]{42}
\][/tex]
Step 4: Calculate [tex]\( f(2.5) \)[/tex]
Now, use the function formula [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(2.5) \)[/tex]:
[tex]\[
f(2.5) = a \cdot b^{2.5}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(2.5) = \left(2 \cdot \sqrt[3]{42}\right) \cdot \left(\sqrt[3]{42}\right)^{2.5}
\][/tex]
Conclusion:
The result for [tex]\( f(2.5) \)[/tex] is approximately 156.61 when rounded to the nearest hundredth.
Step 1: Use the given conditions to create equations
1. [tex]\( f(-1) = 2 \)[/tex] means [tex]\( a \cdot b^{-1} = 2 \)[/tex]. This equation can be rewritten as:
[tex]\[
\frac{a}{b} = 2 \quad \Rightarrow \quad a = 2b
\][/tex]
2. [tex]\( f(2) = 84 \)[/tex] means [tex]\( a \cdot b^2 = 84 \)[/tex].
Step 2: Substitute and solve for [tex]\( b \)[/tex]
Substitute [tex]\( a = 2b \)[/tex] from the first equation into the second equation:
[tex]\[
(2b) \cdot b^2 = 84 \quad \Rightarrow \quad 2b^3 = 84
\][/tex]
Divide both sides by 2:
[tex]\[
b^3 = 42
\][/tex]
Find [tex]\( b \)[/tex] by taking the cube root of 42:
[tex]\[
b = \sqrt[3]{42}
\][/tex]
Step 3: Solve for [tex]\( a \)[/tex] using the value of [tex]\( b \)[/tex]
Since [tex]\( a = 2b \)[/tex], we can substitute the value of [tex]\( b \)[/tex]:
[tex]\[
a = 2 \cdot \sqrt[3]{42}
\][/tex]
Step 4: Calculate [tex]\( f(2.5) \)[/tex]
Now, use the function formula [tex]\( f(x) = a \cdot b^x \)[/tex] to find [tex]\( f(2.5) \)[/tex]:
[tex]\[
f(2.5) = a \cdot b^{2.5}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(2.5) = \left(2 \cdot \sqrt[3]{42}\right) \cdot \left(\sqrt[3]{42}\right)^{2.5}
\][/tex]
Conclusion:
The result for [tex]\( f(2.5) \)[/tex] is approximately 156.61 when rounded to the nearest hundredth.
