The function 𝑓 is given by 𝑓𝑡=𝑒𝑡, and the function 𝑔 is given by 𝑔𝑡=7ln𝑡. If the function ℎ is given by ℎ𝑡=𝑓∘𝑔𝑡, which of the following is an expression for ℎ𝑡, for 𝑡>0 ?Responses7𝑡7 t𝑡7t to the power of 7𝑡𝑒7t e to the power of 7
Question
The function 𝑓 is given by 𝑓𝑡=𝑒𝑡, and the function 𝑔 is given by 𝑔𝑡=7ln𝑡. If the function ℎ is given by ℎ𝑡=𝑓∘𝑔𝑡, which of the following is an expression for ℎ𝑡, for 𝑡>0 ?Responses7𝑡7 t𝑡7t to the power of 7𝑡𝑒7t e to the power of 7
Solution
The function h(t) is given by the composition of the functions f and g, denoted as f(g(t)).
Here, f(t) = e^t and g(t) = 7ln(t).
So, to find h(t), we substitute g(t) into f(t):
h(t) = f(g(t)) = e^(g(t)) = e^(7ln(t)).
Using the property of exponents that a^(bc) = (a^b)^c, we can simplify this to:
h(t) = (e^ln(t))^7.
Since e^ln(t) = t, we have:
h(t) = t^7.
So, the expression for h(t) is t^7.
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