Answer :
Certainly! Let's break down the problem of finding [tex]\( f(-3.5) \)[/tex] for the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex], given [tex]\( f(-5) = 12 \)[/tex] and [tex]\( f(0) = 84 \)[/tex].
### Step 1: Determine [tex]\( a \)[/tex]
Since the function is [tex]\( f(x) = a \cdot b^x \)[/tex], we can find [tex]\( a \)[/tex] using the information that [tex]\( f(0) = 84 \)[/tex].
[tex]\[ f(0) = a \cdot b^0 = a \cdot 1 = a \][/tex]
So, [tex]\( a = 84 \)[/tex].
### Step 2: Find [tex]\( b \)[/tex]
Now that we have [tex]\( a \)[/tex], we use the other given point, [tex]\( f(-5) = 12 \)[/tex].
Setting up the equation:
[tex]\[ f(-5) = a \cdot b^{-5} = 12 \][/tex]
Substitute [tex]\( a = 84 \)[/tex]:
[tex]\[ 84 \cdot b^{-5} = 12 \][/tex]
Solve for [tex]\( b^{-5} \)[/tex]:
[tex]\[ b^{-5} = \frac{12}{84} = \frac{1}{7} \][/tex]
Taking the reciprocal and the fifth root to find [tex]\( b \)[/tex], we have:
[tex]\[ b = \left(\frac{1}{7}\right)^{-\frac{1}{5}} \][/tex]
The solution gives:
[tex]\[ b \approx 1.48 \][/tex]
### Step 3: Calculate [tex]\( f(-3.5) \)[/tex]
Now, with [tex]\( a = 84 \)[/tex] and [tex]\( b \approx 1.48 \)[/tex], we can find [tex]\( f(-3.5) \)[/tex]:
[tex]\[ f(-3.5) = 84 \cdot (1.48)^{-3.5} \][/tex]
From calculations, this results in:
[tex]\[ f(-3.5) \approx 21.51 \][/tex]
Thus, the value of [tex]\( f(-3.5) \)[/tex] is approximately [tex]\( 21.51 \)[/tex], rounded to the nearest hundredth.
### Step 1: Determine [tex]\( a \)[/tex]
Since the function is [tex]\( f(x) = a \cdot b^x \)[/tex], we can find [tex]\( a \)[/tex] using the information that [tex]\( f(0) = 84 \)[/tex].
[tex]\[ f(0) = a \cdot b^0 = a \cdot 1 = a \][/tex]
So, [tex]\( a = 84 \)[/tex].
### Step 2: Find [tex]\( b \)[/tex]
Now that we have [tex]\( a \)[/tex], we use the other given point, [tex]\( f(-5) = 12 \)[/tex].
Setting up the equation:
[tex]\[ f(-5) = a \cdot b^{-5} = 12 \][/tex]
Substitute [tex]\( a = 84 \)[/tex]:
[tex]\[ 84 \cdot b^{-5} = 12 \][/tex]
Solve for [tex]\( b^{-5} \)[/tex]:
[tex]\[ b^{-5} = \frac{12}{84} = \frac{1}{7} \][/tex]
Taking the reciprocal and the fifth root to find [tex]\( b \)[/tex], we have:
[tex]\[ b = \left(\frac{1}{7}\right)^{-\frac{1}{5}} \][/tex]
The solution gives:
[tex]\[ b \approx 1.48 \][/tex]
### Step 3: Calculate [tex]\( f(-3.5) \)[/tex]
Now, with [tex]\( a = 84 \)[/tex] and [tex]\( b \approx 1.48 \)[/tex], we can find [tex]\( f(-3.5) \)[/tex]:
[tex]\[ f(-3.5) = 84 \cdot (1.48)^{-3.5} \][/tex]
From calculations, this results in:
[tex]\[ f(-3.5) \approx 21.51 \][/tex]
Thus, the value of [tex]\( f(-3.5) \)[/tex] is approximately [tex]\( 21.51 \)[/tex], rounded to the nearest hundredth.
