Suppose that the functions $$ f $$ and $$ g $$ are defined as follows. $$ f(x)=\frac{7}{x}, x \neq 0 $$ $$ g(x)=x^{2}-1 $$ Find the compositions $$ f \circ f $$ and $$ g \circ g $$. 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.) $$ (f \circ f)(x)=\square $$ $$ (g \circ g)(x)=\square $$

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UpStudy AI · Accepted Answer

We are given 
$$
f(x)=\frac{7}{x} \quad \text{and} \quad g(x)=x^2-1.
$$

**Step 1. Find $$(f \circ f)(x)$$.**

By definition, $$(f \circ f)(x)=f(f(x))$$.

Substitute $$f(x)$$ into $$f$$:
$$
f(f(x))=f\left(\frac{7}{x}\right)=\frac{7}{\frac{7}{x}}.
$$

Simplify the expression:
$$
\frac{7}{\frac{7}{x}} = \frac{7x}{7} = x.
$$

Thus,
$$
(f \circ f)(x)=x.
$$

**Step 2. Find $$(g \circ g)(x)$$.**

By definition, $$(g \circ g)(x)=g(g(x))$$.

Substitute $$g(x)$$ into $$g$$:
$$
g(g(x)) = g(x^2-1) = (x^2-1)^2 - 1.
$$

Now, expand $$(x^2-1)^2$$:
$$
(x^2-1)^2 = x^4 - 2x^2 + 1.
$$

Subtract 1:
$$
g(g(x)) = x^4 - 2x^2 + 1 - 1 = x^4 - 2x^2.
$$

Thus,
$$
(g \circ g)(x)=x^4-2x^2.
$$

**Final Answers:**

$$
(f \circ f)(x)=x, \quad (g \circ g)(x)=x^4-2x^2.
$$
$$ (f \circ f)(x) = x, \quad (g \circ g)(x) = x^4 - 2x^2 $$

Suppose that the functions and are defined as follows.

Find the compositions and .
Simplify. your answers as much as possible.
(Assume that your expressions are defined for all in the domain of the composition. You do not have to indicate the domain.)

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Suppose that the functions and are defined as follows. Find the compositions and . Simplify. your answers as much as possible. (Assume that your expressions are defined for all in the domain of the composition. You do not have to indicate the domain.)