Pregunta
Given: \( g(x)=\sqrt{x-4} \) and \( h(x)=2 x-8 \). What are the restrictions on the domain of \( g \circ h \) ? \( x \geq \square \) DONE:
Ask by Chan Hammond. in the United States
Mar 21,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The domain restriction for \( g \circ h \) is \( x \geq 6 \).
Solución
1. The composition \( g \circ h \) means we need to consider \( g(h(x)) \).
2. Since \( g(x) = \sqrt{x-4} \), the radicand must be nonnegative. This gives the restriction:
\[
h(x)-4 \geq 0
\]
3. Substitute \( h(x) = 2x-8 \) into the inequality:
\[
2x-8-4 \geq 0
\]
4. Simplify the inequality:
\[
2x - 12 \geq 0
\]
5. Solve for \( x \):
\[
2x \geq 12 \quad \Rightarrow \quad x \geq 6
\]
Thus, the domain restriction for \( g \circ h \) is \( x \geq 6 \).
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Simplifique esta solución The Deep Dive
To find the restrictions on the domain of \( g \circ h \), we first need to determine the input for \( g \) based on the output of \( h \). The function \( h(x) = 2x - 8 \) needs to be within the valid domain of \( g(x) = \sqrt{x - 4} \). For \( g(x) \) to be defined, we require \( x - 4 \geq 0 \), which simplifies to \( x \geq 4 \). Now, substituting \( h(x) \) into this inequality, we have \( h(x) \geq 4 \): \[ 2x - 8 \geq 4 \] Solving for \( x \), we get: \[ 2x \geq 12 \implies x \geq 6. \] Thus, for \( g \circ h \) to be defined, we need \( x \geq 6 \). So, the restriction on the domain of \( g \circ h \) is \( x \geq 6 \).
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