Suppose that the functions \( g \) and \( h \) are defined as follows. \[ g(x)=x^{2}+7 \] \( h(x)=\frac{8}{9 x}, x \neq 0 \) Find the compositions \( g \circ g \) and \( h \) oh. Simplify, your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate the domain.) \( (g \circ g)(x)=\square \)

We first compute the composition \( (g\circ g)(x) \): 1. Write out the composition: \[ (g \circ g)(x)=g(g(x)) \] 2. Substitute \( g(x)=x^2+7 \): \[ g(g(x))=g(x^2+7) \] 3. Replace \( x \) by \( x^2+7 \) in the definition of \( g \): \[ g(x^2+7)=(x^2+7)^2+7 \] 4. Expand \( (x^2+7)^2 \) (if desired): \[ (x^2+7)^2=x^4+14x^2+49 \] 5. Add 7 to obtain: \[ (g\circ g)(x)=x^4+14x^2+49+7=x^4+14x^2+56 \] The final simplified form is: \[ (g\circ g)(x)=x^4+14x^2+56 \] Next, we compute the composition \( (h\circ h)(x) \) (assuming "h oh" means \( h \circ h \)): 1. Write out the composition: \[ (h \circ h)(x)=h(h(x)) \] 2. Substitute \( h(x)=\frac{8}{9x} \): \[ h\left(\frac{8}{9x}\right)=\frac{8}{9\left(\frac{8}{9x}\right)} \] 3. Simplify the expression: \[ \frac{8}{9\left(\frac{8}{9x}\right)}=\frac{8}{\frac{8}{x}}=\frac{8x}{8}=x \] The final simplified form is: \[ (h\circ h)(x)=x \]