To rewrite the function \( f(x) = 7 \cdot 2^{2x + 3} \cdot 2^{-(x + 5)} + 3 \) in the form \( f(x) = a(b)^{x - h} + k \), we need to simplify the expression using the properties of exponents.
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Notice that \( 2^{2x + 3} \cdot 2^{-(x + 5)} \) can be combined using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \): \[ 2^{2x + 3} \cdot 2^{-(x + 5)} = 2^{(2x + 3) - (x + 5)} = 2^{2x + 3 - x - 5} = 2^{x - 2} \]
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Substitute this back into the function: \[ f(x) = 7 \cdot 2^{x - 2} + 3 \]
Now we can identify the values for \(a\), \(b\), \(h\), and \(k\):
- \( a = 7 \)
- \( b = 2 \)
- \( h = 2 \) (since it is \( x - 2 \))
- \( k = 3 \)
So the function \( f(x) = 7 \cdot 2^{x - 2} + 3 \) is in the desired form.
From the provided options, the correct rewritten function is:
f(x)=7⋅2x−2+3
This corresponds to the third option from your list.