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Answer :
To solve the problem, we're given that [tex]\( f(x) = a \cdot b^x \)[/tex] is an exponential function, and we're asked to find the value of [tex]\( f(-0.5) \)[/tex]. Let's use the information provided about the values at certain points to determine the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex], and then find [tex]\( f(-0.5) \)[/tex].
### Step 1: Determine the value of [tex]\( a \)[/tex]
The function [tex]\( f(x) = a \cdot b^x \)[/tex] evaluates to 82 when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = a \cdot b^0 = a \][/tex]
Since [tex]\( f(0) = 82 \)[/tex], we find that:
[tex]\[ a = 82 \][/tex]
### Step 2: Use another point to find [tex]\( b \)[/tex]
We are given another point where [tex]\( f(-3.5) = 6 \)[/tex].
[tex]\[ f(-3.5) = a \cdot b^{-3.5} = 6 \][/tex]
Substituting [tex]\( a = 82 \)[/tex] into this equation, we have:
[tex]\[ 82 \cdot b^{-3.5} = 6 \][/tex]
To solve for [tex]\( b \)[/tex], divide both sides by 82:
[tex]\[ b^{-3.5} = \frac{6}{82} \][/tex]
Now, take both sides to the power of [tex]\( -\frac{1}{3.5} \)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \left(\frac{6}{82}\right)^{-\frac{1}{3.5}} \][/tex]
### Step 3: Calculate [tex]\( f(-0.5) \)[/tex]
Now that we have [tex]\( a = 82 \)[/tex] and we've calculated [tex]\( b \)[/tex], we need to determine [tex]\( f(-0.5) \)[/tex].
[tex]\[ f(-0.5) = 82 \cdot b^{-0.5} \][/tex]
Substitute [tex]\( b \)[/tex] into this equation to find [tex]\( f(-0.5) \)[/tex]. Using the calculated [tex]\( b \)[/tex]:
[tex]\[ f(-0.5) \approx 56.44 \][/tex]
Thus, the value of [tex]\( f(-0.5) \)[/tex], rounded to the nearest hundredth, is approximately 56.44.
### Step 1: Determine the value of [tex]\( a \)[/tex]
The function [tex]\( f(x) = a \cdot b^x \)[/tex] evaluates to 82 when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = a \cdot b^0 = a \][/tex]
Since [tex]\( f(0) = 82 \)[/tex], we find that:
[tex]\[ a = 82 \][/tex]
### Step 2: Use another point to find [tex]\( b \)[/tex]
We are given another point where [tex]\( f(-3.5) = 6 \)[/tex].
[tex]\[ f(-3.5) = a \cdot b^{-3.5} = 6 \][/tex]
Substituting [tex]\( a = 82 \)[/tex] into this equation, we have:
[tex]\[ 82 \cdot b^{-3.5} = 6 \][/tex]
To solve for [tex]\( b \)[/tex], divide both sides by 82:
[tex]\[ b^{-3.5} = \frac{6}{82} \][/tex]
Now, take both sides to the power of [tex]\( -\frac{1}{3.5} \)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \left(\frac{6}{82}\right)^{-\frac{1}{3.5}} \][/tex]
### Step 3: Calculate [tex]\( f(-0.5) \)[/tex]
Now that we have [tex]\( a = 82 \)[/tex] and we've calculated [tex]\( b \)[/tex], we need to determine [tex]\( f(-0.5) \)[/tex].
[tex]\[ f(-0.5) = 82 \cdot b^{-0.5} \][/tex]
Substitute [tex]\( b \)[/tex] into this equation to find [tex]\( f(-0.5) \)[/tex]. Using the calculated [tex]\( b \)[/tex]:
[tex]\[ f(-0.5) \approx 56.44 \][/tex]
Thus, the value of [tex]\( f(-0.5) \)[/tex], rounded to the nearest hundredth, is approximately 56.44.
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