Questions: h(n)=3 n+5, g(n)=2 n-5, Find (h ∘ g)(-3 z)

Solution Steps

To solve the problem, we need to find the composition of the functions \( h(n) \) and \( g(n) \), denoted as \( (h \circ g)(n) \). This means we first apply \( g(n) \) and then apply \( h \) to the result of \( g(n) \). After finding the expression for \( (h \circ g)(n) \), we substitute \( n = -3z \) into this expression to find the final result.

We have two functions defined as follows: \[ h(n) = 3n + 5 \] \[ g(n) = 2n - 5 \]

To find the composition \( (h \circ g)(n) \), we first compute \( g(n) \): \[ g(n) = 2n - 5 \] Next, we substitute \( g(n) \) into \( h(n) \): \[ (h \circ g)(n) = h(g(n)) = h(2n - 5) = 3(2n - 5) + 5 \] Simplifying this expression: \[ = 6n - 15 + 5 = 6n - 10 \]

Now, we substitute \( n = -3z \) into the composition: \[ (h \circ g)(-3z) = 6(-3z) - 10 = -18z - 10 \] For \( z = 1 \): \[ (h \circ g)(-3 \cdot 1) = -18 \cdot 1 - 10 = -18 - 10 = -28 \]

Final Answer