WEBVTT
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Consider the function 𝑓 of 𝑥 is equal to 𝑏 to the power of 𝑥, where 𝑏 is a positive real number not equal to one.
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What is the domain of the inverse of 𝑓 of 𝑥?
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There are a few ways of approaching this problem.
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One way would be to recall that exponential functions and logarithmic functions are the inverse of each other.
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This means that if 𝑓 of 𝑥 is equal to 𝑏 to the power of 𝑥, the inverse function is equal to log base 𝑏 of 𝑥.
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We are asked to find the domain of this function.
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The domain of any function is the set of input values.
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We know that we can only find the logarithm of positive values.
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This means that the domain of the inverse function is 𝑥 is greater than zero as the only values we can substitute into the function log base 𝑏 of 𝑥 are 𝑥 greater than zero.
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An alternative method here would be to consider the graphs of our functions.
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The graph of 𝑓 of 𝑥 is shown.
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It intersects the 𝑦-axis at 𝑏 and the 𝑥-axis is an asymptote.
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The inverse of any function is its reflection in the line 𝑦 equals 𝑥.
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This means that the function log base 𝑏 of 𝑥 intersects the 𝑥-axis at 𝑏 and the 𝑦-axis is an asymptote.
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As the domain is the set of input values, we can see from the graph that the domain of the inverse of 𝑓 of 𝑥 is all numbers greater than zero.
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A final method would be to recall that the domain of 𝑓 is equal to the range of the inverse.
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Likewise, the range of 𝑓 of 𝑥 is equal to the domain of the inverse.
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The range of any function is the set of output values.
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We can see from the graph that the range of 𝑓 of 𝑥 is all values greater than zero.
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This once again proves that the domain of the inverse function is 𝑥 is greater than zero.