Pergunta

D log_(4)x](https://static.questionai.br.com/resource%2Fqaiseoimg%2F202501%2Ffunction-f-increasing-function-every-time-output-values-tTZTCrYwAN0q.jpg?x-oss-process=image/resize,w_558,h_500/quality,q_35/format,webp)
The function f is an increasing function such that every time the output values of the function f increase by 1, the corresponding input values multiply by 4. Which of the following could define f(x) A x^4 B 4^x C sqrt [4](x) D log_(4)x
Solução

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AlexandreMestre · Tutor por 5 anos
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To determine which function \( f(x) \) satisfies the given condition, we need to analyze the behavior of each option when the input values are multiplied by 4.<br /><br />Given:<br />- The function \( f \) is an increasing function.<br />- Every time the output values of the function \( f \) increase by 1, the corresponding input values multiply by 4.<br /><br />Let's analyze each option:<br /><br />A. \( f(x) = x^4 \)<br /> - If \( x \) is multiplied by 4, then \( (4x)^4 = 4^4 \cdot x^4 = 256x^4 \). This does not increase the output by 1.<br /><br />B. \( f(x) = 4^x \)<br /> - If \( x \) is multiplied by 4, then \( f(4x) = 4^{4x} = (4^x)^4 = (f(x))^4 \). This does not increase the output by 1.<br /><br />C. \( f(x) = \sqrt[4]{x} \)<br /> - If \( x \) is multiplied by 4, then \( f(4x) = \sqrt[4]{4x} = \sqrt[4]{4} \cdot \sqrt[4]{x} = \sqrt[4]{4} \cdot f(x) \). This does not increase the output by 1.<br /><br />D. \( f(x) = \log_4(x) \)<br /> - If \( x \) is multiplied by 4, then \( f(4x) = \log_4(4x) = \log_4(4) + \log_4(x) = 1 + \log_4(x) = 1 + f(x) \). This increases the output by 1.<br /><br />Therefore, the correct answer is:<br />D. \( \log_4(x) \)
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