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If 𝑔 of 𝑥 is the inverse of the function 𝑓 of 𝑥, which is equal to two to the power of 𝑥 plus five, find 𝑔 of 𝑥.
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We begin by recalling that the logarithmic function is the inverse of an exponential function.
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If 𝑓 of 𝑥 is equal to 𝑎 to the power of 𝑥, then the inverse of 𝑓 of 𝑥 is equal to log base 𝑎 of 𝑥 where 𝑎 is positive and not equal to one.
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In this question, we are told that 𝑓 of 𝑥 is equal to two to the power of 𝑥 plus five.
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This can be rewritten as 𝑦 is equal to two to the power of 𝑥 plus five.
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To find the inverse of any function, we exchange the variables 𝑥 and 𝑦 and then solve for 𝑦.
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We begin with the equation 𝑥 is equal to two to the power of 𝑦 plus five.
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Using the fact that a logarithmic function is the inverse of an exponential function, this can be rewritten as 𝑦 plus five is equal to log base two of 𝑥.
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We can then subtract five from both sides of this equation so that 𝑦 is equal to log base two of 𝑥 minus five.
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We then replace 𝑦 with the inverse function 𝑔 of 𝑥.
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The inverse of two the power of 𝑥 plus five is log base two of 𝑥 minus five.