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Function transformation involves changing the form of a function
The function g(x) is [tex]\mathbf{g(x) = 8(2)^x}[/tex]
The function is given as:
[tex]\mathbf{f(x) = 3^x}[/tex]
g(x) is an exponential function that passes through points (-2,2) and (-1,4).
An exponential function is represented as:
[tex]\mathbf{y = ab^x}[/tex]
At point (-2,2), we have:
[tex]\mathbf{2 = ab^{-2}}[/tex]
At point (-1,4), we have:
[tex]\mathbf{4 = ab^{-1}}[/tex]
Divide both equations
[tex]\mathbf{\frac 42=\frac{ab^{-1}}{ab^{-2}}}[/tex]
Simplify
[tex]\mathbf{2=\frac{b^{-1}}{b^{-2}}}[/tex]
Apply law of indices
[tex]\mathbf{2=b^{-1+2}}[/tex]
[tex]\mathbf{2=b}[/tex]
Rewrite as:
[tex]\mathbf{b =2}[/tex]
Substitute 2 for b in [tex]\mathbf{2 = ab^{-2}}[/tex]
[tex]\mathbf{2 =a(2^{-2})}[/tex]
This gives
[tex]\mathbf{2 =a(\frac 14)}[/tex]
Multiply both sides by 4
[tex]\mathbf{a = 8}[/tex]
Substitute 8 for (a) and 2 for (b) in [tex]\mathbf{y = ab^x}[/tex]
[tex]\mathbf{y = 8(2)^x}[/tex]
Express as a function
[tex]\mathbf{g(x) = 8(2)^x}[/tex]
Hence, the function g(x) is [tex]\mathbf{g(x) = 8(2)^x}[/tex]
Read more about exponential functions at:
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