Pregunta
State the domain and range of the function. \( f(x)=\log _{5}(x+2) \) Domain: Range:
Ask by Chan Pritchard. in the United States
Mar 12,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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Domain: \( x > -2 \)
Range: All real numbers
Solución
1. The function given is
\[
f(x)=\log_5(x+2).
\]
2. For a logarithmic function \( \log_b(u) \), the argument \( u \) must be positive. Therefore, we set:
\[
x+2 > 0.
\]
3. Solving for \( x \):
\[
x > -2.
\]
4. Hence, the domain of \( f(x) \) is:
\[
(-2, \infty).
\]
5. The range of any logarithmic function is all real numbers, therefore:
\[
(-\infty, \infty).
\]
Domain: \( (-2, \infty) \)
Range: \( (-\infty, \infty) \)
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Simplifique esta solución Mind Expander
The domain of the function \( f(x) = \log_{5}(x+2) \) consists of all values \( x \) for which the expression inside the logarithm is positive. Since \( x + 2 > 0 \), we have \( x > -2 \). Therefore, the domain is \( (-2, \infty) \). As for the range, logarithmic functions can produce all real numbers as outputs. Thus, regardless of the base, \( f(x) \) can take any value from negative infinity to positive infinity. Hence, the range is \( (-\infty, \infty) \).
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