Let #h(t) = 1/(t^2)# and #g(t)= sqrt(3t+5)#, how do you find each of the compositions and domain and range?

Question

Elijah Allen · Accepted Answer

undefined To find the compositions of $ h(t) \circ g(t) $ and $ g(t) \circ h(t) $, and their respective domains and ranges:

1. $ h(t) \circ g(t) $:
   - Substitute $ g(t) $ into $ h(t) $, i.e., $ h(g(t)) = h(\sqrt{3t+5}) $.
   - Find the domain of $ h(g(t)) $ by considering the domain of $ g(t) $ and ensuring it satisfies the domain of $ h(t) $.
   - Determine the range of $ h(g(t)) $ by analyzing the range of $ g(t) $ and applying $ h(t) $.

2. $ g(t) \circ h(t) $:
   - Substitute $ h(t) $ into $ g(t) $, i.e., $ g(h(t)) = g\left(\frac{1}{t^2}\right) $.
   - Find the domain of $ g(h(t)) $ by considering the domain of $ h(t) $ and ensuring it satisfies the domain of $ g(t) $.
   - Determine the range of $ g(h(t)) $ by analyzing the range of $ h(t) $ and applying $ g(t) $.

For $ h(t) \circ g(t) $, the domain and range will depend on the properties of $ h(t) $ and $ g(t) $, and similarly for $ g(t) \circ h(t) $.

Cora Alexander · Answer

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