1. Refer to the function $g(x) = 7x + 2$:

(a) Domain and Range of $g$

• Domain: $g(x) = 7x + 2$ is a linear function, which is defined for all real numbers. Thus, the domain is $(-\infty, \infty)$.
• Range: Since $g(x)$ is linear and covers all real numbers, its range is also $(-\infty, \infty)$.

(b) Algebraic Expression for $g(h(x))$

Given $h(x) = x^2 + x + 1$, substitute $h(x)$ into $g(x)$: $g(h(x)) = 7h(x) + 2 = 7(x^2 + x + 1) + 2 = 7x^2 + 7x + 7 + 2 = 7x^2 + 7x + 9$

(c) Algebraic Expression for $3g(x) - h(x)$

Substitute $g(x) = 7x + 2$ and $h(x) = x^2 + x + 1$: $3g(x) - h(x) = 3(7x + 2) - (x^2 + x + 1) = 21x + 6 - x^2 - x - 1 = -x^2 + 20x + 5$

(d) Function Equation for the Inverse, $g^{-1}(x)$

To find the inverse, solve $y = 7x + 2$ for $x$: $y = 7x + 2 \implies x = \frac{y - 2}{7}$ Thus, the inverse function is: $g^{-1}(x) = \frac{x - 2}{7}$

(e) Domain and Range of $g^{-1}$

• Domain: Since $g^{-1}(x) = \frac{x - 2}{7}$ is defined for all $x$, its domain is $(-\infty, \infty)$.
• Range: The output of $g^{-1}(x)$ is also all real numbers, so the range is $(-\infty, \infty)$.

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2. Refer to the graph of $R(x)$:

(a) Domain and Range of $R$

From the given points $(-7.5, 0), (-1, 2), (0, 4.5), (6, 6)$:

• Domain: The set of $x$-values is $\{-7.5, -1, 0, 6\}$.
• Range: The set of $y$-values is $\{0, 2, 4.5, 6\}$.

(b) Graph of the Inverse, $R^{-1}$

To graph $R^{-1}$, reflect each point of $R(x)$ over the line $y = x$. The new points are: $R^{-1} = \{(0, -7.5), (2, -1), (4.5, 0), (6, 6)\}$

(c) Domain and Range of $R^{-1}$

• Domain: The set of $x$-values of $R^{-1}$ (which is the range of $R$) is $\{0, 2, 4.5, 6\}$.
• Range: The set of $y$-values of $R^{-1}$ (which is the domain of $R$) is $\{-7.5, -1, 0, 6\}$.

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3. Refer to $f$ in Table 3.14:

$x$	$f(x)$
0	3
1	0
2	2
3	-4

(a) Domain and Range of $f$

• Domain: The set of $x$-values is $\{0, 1, 2, 3\}$.
• Range: The set of $y$-values is $\{3, 0, 2, -4\}$.

(b) Function Table for $f^{-1}$

The inverse function swaps the $x$- and $y$-values of $f$:

$x$	$f^{-1}(x)$
3	0
0	1
2	2
-4	3

(c) Domain and Range of $f^{-1}$

• Domain: The set of $x$-values of $f^{-1}$ (which is the range of $f$) is $\{3, 0, 2, -4\}$.
• Range: The set of $y$-values of $f^{-1}$ (which is the domain of $f$) is $\{0, 1, 2, 3\}$.

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Additional Questions:

1. How does the graph of $g(x)$ relate to its inverse $g^{-1}(x)$?
2. What happens if you compose $g$ with $g^{-1}$?
3. Can you generalize the inverse formula for any linear function $ax + b$?
4. How do the properties of the graph $R(x)$ change when reflected over $y = x$?
5. What insights can be drawn by analyzing $f(x)$ and $f^{-1}(x)$ in terms of their domains and ranges?

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Tip:

For any function $f$, its inverse $f^{-1}$ exists if and only if $f$ is one-to-one (passes the horizontal line test).