In the exercises, from 1 to 6, find and graph the inverse.
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\[ f(x) = 2x+1 \]
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\[ g(x) = x^2-1, \; x \geq 0 \]
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\[ h(x) = x^3 + 2 \]
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\[ k(x) = \frac{1}{x} -1 \]
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\[ f(x) = \sqrt{16-2x} \]
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\[ g(x) = \frac{5x-15}{3x+7} \]
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Formally prove that:
- If \(f\) is increasing, then \(f^{-1}\) is increasing.
- If \(f\) is decreasing, then \(f^{-1}\) is decreasing.
Answers
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\(f^{-1} (x) = \cfrac{1}{2}x – \cfrac{1}{2}\)
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\(g^{-1} (x) = \sqrt{ x + 1 }\)
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\(h^{-1} (x) = \sqrt[3]{ x – 2 }\)
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\(k^{-1} (x) = \cfrac{1}{x + 1}\)
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\(f^{-1} (x) = 8 – \cfrac{x^2}{2}\)
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\(g^{-1} (x) = \cfrac{ -7x – 15 }{ 3x – 5 }\)
