Answer :
To solve the problem where [tex]\( f(x) \)[/tex] is an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex], and we are given that [tex]\( f(-5) = 12 \)[/tex] and [tex]\( f(0) = 84 \)[/tex], we need to find the value of [tex]\( f(-3.5) \)[/tex].
### Step 1: Determine the value of [tex]\( a \)[/tex]
Since [tex]\( f(0) = 84 \)[/tex], substituting into the exponential function gives us:
[tex]\[
f(0) = a \cdot b^0 = a \cdot 1 = a
\][/tex]
So, [tex]\( a = 84 \)[/tex].
### Step 2: Find the base [tex]\( b \)[/tex]
We know that [tex]\( f(-5) = 12 \)[/tex], so:
[tex]\[
f(-5) = a \cdot b^{-5} = 12
\][/tex]
Substituting [tex]\( a = 84 \)[/tex] into the equation:
[tex]\[
84 \cdot b^{-5} = 12
\][/tex]
To solve for [tex]\( b^{-5} \)[/tex]:
[tex]\[
b^{-5} = \frac{12}{84}
\][/tex]
Simplifying:
[tex]\[
b^{-5} = \frac{1}{7}
\][/tex]
Thus, raising both sides to the power of [tex]\(-1/5\)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[
b^5 = 7
\][/tex]
Taking the fifth root:
[tex]\[
b = 7^{1/5} \approx 1.48
\][/tex]
### Step 3: Calculate [tex]\( f(-3.5) \)[/tex]
Now, we want to find [tex]\( f(-3.5) \)[/tex]:
[tex]\[
f(-3.5) = a \cdot b^{-3.5}
\][/tex]
Substituting the known values [tex]\( a = 84 \)[/tex] and [tex]\( b \approx 1.48 \)[/tex]:
[tex]\[
f(-3.5) \approx 84 \cdot (1.48)^{-3.5}
\][/tex]
Carrying out the calculation:
[tex]\[
f(-3.5) \approx 21.51
\][/tex]
Thus, the value of [tex]\( f(-3.5) \)[/tex], rounded to the nearest hundredth, is approximately [tex]\( 21.51 \)[/tex].
### Step 1: Determine the value of [tex]\( a \)[/tex]
Since [tex]\( f(0) = 84 \)[/tex], substituting into the exponential function gives us:
[tex]\[
f(0) = a \cdot b^0 = a \cdot 1 = a
\][/tex]
So, [tex]\( a = 84 \)[/tex].
### Step 2: Find the base [tex]\( b \)[/tex]
We know that [tex]\( f(-5) = 12 \)[/tex], so:
[tex]\[
f(-5) = a \cdot b^{-5} = 12
\][/tex]
Substituting [tex]\( a = 84 \)[/tex] into the equation:
[tex]\[
84 \cdot b^{-5} = 12
\][/tex]
To solve for [tex]\( b^{-5} \)[/tex]:
[tex]\[
b^{-5} = \frac{12}{84}
\][/tex]
Simplifying:
[tex]\[
b^{-5} = \frac{1}{7}
\][/tex]
Thus, raising both sides to the power of [tex]\(-1/5\)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[
b^5 = 7
\][/tex]
Taking the fifth root:
[tex]\[
b = 7^{1/5} \approx 1.48
\][/tex]
### Step 3: Calculate [tex]\( f(-3.5) \)[/tex]
Now, we want to find [tex]\( f(-3.5) \)[/tex]:
[tex]\[
f(-3.5) = a \cdot b^{-3.5}
\][/tex]
Substituting the known values [tex]\( a = 84 \)[/tex] and [tex]\( b \approx 1.48 \)[/tex]:
[tex]\[
f(-3.5) \approx 84 \cdot (1.48)^{-3.5}
\][/tex]
Carrying out the calculation:
[tex]\[
f(-3.5) \approx 21.51
\][/tex]
Thus, the value of [tex]\( f(-3.5) \)[/tex], rounded to the nearest hundredth, is approximately [tex]\( 21.51 \)[/tex].
