Answer :
To find the value of [tex]\( f(-2) \)[/tex] for the exponential function [tex]\( y = a \cdot b^x \)[/tex], given that [tex]\( f(-4.5) = 21 \)[/tex] and [tex]\( f(0.5) = 82 \)[/tex], we can use the following steps:
1. Set Up the Equations:
We know:
[tex]\[
a \cdot b^{-4.5} = 21
\][/tex]
[tex]\[
a \cdot b^{0.5} = 82
\][/tex]
2. Find the Base [tex]\( b \)[/tex]:
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{0.5}}{a \cdot b^{-4.5}} = \frac{82}{21}
\][/tex]
Simplify the equation:
[tex]\[
b^{0.5 + 4.5} = \frac{82}{21}
\][/tex]
[tex]\[
b^5 = \frac{82}{21}
\][/tex]
Solve for [tex]\( b \)[/tex] by taking the fifth root:
[tex]\[
b = \left( \frac{82}{21} \right)^{1/5}
\][/tex]
3. Find the Coefficient [tex]\( a \)[/tex]:
Substitute the value of [tex]\( b \)[/tex] back into one of the equations, for instance, [tex]\( a \cdot b^{0.5} = 82 \)[/tex]:
[tex]\[
a = \frac{82}{b^{0.5}}
\][/tex]
4. Calculate [tex]\( f(-2) \)[/tex]:
Now, use the expression for the function to find [tex]\( f(-2) \)[/tex]:
[tex]\[
f(-2) = a \cdot b^{-2}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] calculated earlier:
[tex]\[
f(-2) = \frac{82}{b^{0.5}} \cdot b^{-2}
\][/tex]
5. Final Result:
Compute the value using the calculated [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(-2) \approx 41.50
\][/tex]
So, the value of [tex]\( f(-2) \)[/tex] is approximately [tex]\( 41.50 \)[/tex] to the nearest hundredth.
1. Set Up the Equations:
We know:
[tex]\[
a \cdot b^{-4.5} = 21
\][/tex]
[tex]\[
a \cdot b^{0.5} = 82
\][/tex]
2. Find the Base [tex]\( b \)[/tex]:
Divide the second equation by the first to eliminate [tex]\( a \)[/tex]:
[tex]\[
\frac{a \cdot b^{0.5}}{a \cdot b^{-4.5}} = \frac{82}{21}
\][/tex]
Simplify the equation:
[tex]\[
b^{0.5 + 4.5} = \frac{82}{21}
\][/tex]
[tex]\[
b^5 = \frac{82}{21}
\][/tex]
Solve for [tex]\( b \)[/tex] by taking the fifth root:
[tex]\[
b = \left( \frac{82}{21} \right)^{1/5}
\][/tex]
3. Find the Coefficient [tex]\( a \)[/tex]:
Substitute the value of [tex]\( b \)[/tex] back into one of the equations, for instance, [tex]\( a \cdot b^{0.5} = 82 \)[/tex]:
[tex]\[
a = \frac{82}{b^{0.5}}
\][/tex]
4. Calculate [tex]\( f(-2) \)[/tex]:
Now, use the expression for the function to find [tex]\( f(-2) \)[/tex]:
[tex]\[
f(-2) = a \cdot b^{-2}
\][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] calculated earlier:
[tex]\[
f(-2) = \frac{82}{b^{0.5}} \cdot b^{-2}
\][/tex]
5. Final Result:
Compute the value using the calculated [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
f(-2) \approx 41.50
\][/tex]
So, the value of [tex]\( f(-2) \)[/tex] is approximately [tex]\( 41.50 \)[/tex] to the nearest hundredth.
