1. Refer to the function g(x) = 7x + 2.
(a) What are the domain and range of g?
(b) Let h(x) = x2 + x + 1. Find an algebraic expression for g(h(x)).
(c) Again using h(x) = x2 + x + 1, find an algebraic expression for 3g(x)  h(x).
(d) Find a function equation for the inverse function, g 1
.
(e) What are the domain and range of g 1?
2. Refer to the graph of R(x) in Figure 3.2. The diagonal line y= x is included on the graph for
reference. Note that the graph of R(x) includes the points ( 7.5, 0), ( 1, 2), (0, 4.5), and (6, 6).
(a) What are the domain and range of R?
(b) Draw the graph of the inverse function, R 1
.
(c) What are the domain and range of R 1?
Figure 3.2: Graph of R(x)
x 0 1 2 3
f (x) 3 0 2 -4
Table 3.14: f (x)
3. Refer to f in Table 3.14.
(a) What are the domain and range of f ?
(b) Find a function table for the inverse function, f 1
.
(c) What are the domain and range of f 1?

### Problem 1: Refer to \( g(x) = 7x + 2 \)
---

#### (a) **Domain and Range of \( g(x) \):**

- **Domain**: The domain of \( g(x) \) is all real numbers (\( \mathbb{R} \)), since there are no restrictions on \( x \).
- **Range**: The function \( g(x) \) is a linear function with a non-zero slope (\( 7 \)), so its range is also all real numbers (\( \mathbb{R} \)).

#### (b) **Algebraic Expression for \( g(h(x)) \):**
Let \( h(x) = x^2 + x + 1 \). Substituting \( h(x) \) 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 \( 3g(x) \circ h(x) \):**
We need \( 3g(x) \circ h(x) \), which implies:
\[
3g(x) \circ h(x) = 3g(h(x)) = 3(7x^2 + 7x + 9) = 21x^2 + 21x + 27
\]

#### (d) **Function Equation for the Inverse Function \( g^{-1}(x) \):**
To find \( g^{-1}(x) \), solve for \( x \) in \( y = 7x + 2 \):
\[
y = 7x + 2 \implies y - 2 = 7x \implies x = \frac{y - 2}{7}
\]
Thus, the inverse function is:
\[
g^{-1}(x) = \frac{x - 2}{7}
\]

#### (e) **Domain and Range of \( g^{-1}(x) \):**
- **Domain**: Since the range of \( g(x) \) is all real numbers, the domain of \( g^{-1}(x) \) is also \( \mathbb{R} \).
- **Range**: The range of \( g^{-1}(x) \) is the domain of \( g(x) \), which is \( \mathbb{R} \).

---

### Problem 2: Refer to the graph of \( R(x) \)
---

#### (a) **Domain and Range of \( R(x) \):**
From the graph, the \( x \)-values span from \( -7.5 \) to \( 6 \), and the \( y \)-values span from \( 0 \) to \( 6 \).

- **Domain**: \( [-7.5, 6] \)
- **Range**: \( [0, 6] \)

#### (b) **Graph of the Inverse Function \( R^{-1}(x) \):**
The inverse function graph \( R^{-1}(x) \) is obtained by reflecting the points of \( R(x) \) across the line \( y = x \). The points for \( R(x) \) become:
\[
(-7.5, 0) \rightarrow (0, -7.5), \, (-1, 2) \rightarrow (2, -1), \, (0, 4.5) \rightarrow (4.5, 0), \, (6, 6) \rightarrow (6, 6)
\]
Plot these points to obtain \( R^{-1}(x) \).

#### (c) **Domain and Range of \( R^{-1}(x) \):**
- **Domain**: Since the range of \( R(x) \) is \( [0, 6] \), the domain of \( R^{-1}(x) \) is \( [0, 6] \).
- **Range**: Since the domain of \( R(x) \) is \( [-7.5, 6] \), the range of \( R^{-1}(x) \) is \( [-7.5, 6] \).

---

### Problem 3: Refer to \( f(x) \) in Table 3.14
---

#### (a) **Domain and Range of \( f(x) \):**
From the table:
- **Domain**: The \( x \)-values are \( \{0, 1, 2, 3\} \).
- **Range**: The \( f(x) \)-values are \( \{3, 0, 2, -4\} \).

#### (b) **Function Table for the Inverse Function \( f^{-1}(x) \):**
To find \( f^{-1}(x) \), swap the domain and range of \( f(x) \):
\[
f(x) = \{ (0, 3), (1, 0), (2, 2), (3, -4) \} \implies f^{-1}(x) = \{ (3, 0), (0, 1), (2, 2), (-4, 3) \}
\]

#### (c) **Domain and Range of \( f^{-1}(x) \):**
- **Domain**: The range of \( f(x) \), which is \( \{3, 0, 2, -4\} \).
- **Range**: The domain of \( f(x) \), which is \( \{0, 1, 2, 3\} \).

---

### Follow-Up Questions:
1. How can we verify the correctness of the inverse functions derived in each case?
2. What is the relationship between the slopes of a linear function and its inverse?
3. How does the reflection property \( y = x \) affect the graphical representation of an inverse function?
4. Can the composition of \( g \) and \( g^{-1} \) demonstrate their inverse relationship? Show examples.
5. What happens to the domain and range if \( g(x) = 7x + 2 \) were modified to a quadratic function?

### Tip:
To check if two functions are inverses, calculate their compositions (\( f(g(x)) \) and \( g(f(x)) \)); if both equal \( x \), they are inverses.

1. Refer to the function g(x) = 7x + 2.
(a) What are the domain and range of g?
(b) Let h(x) = x^2 + x + 1. Find an algebraic expression for g(h(x)).
(c) Again using h(x) = x^2 + x + 1, find an algebraic expression for 3g(x) ∘ h(x).
(d) Find a function equation for the inverse function, g^{-1}.
(e) What are the domain and range of g^{-1}?
2. Refer to the graph of R(x) in Figure 3.2. The diagonal line y = x is included on the graph for reference. Note that the graph of R(x) includes the points (-7.5, 0), (-1, 2), (0, 4.5), and (6, 6).
(a) What are the domain and range of R?
(b) Draw the graph of the inverse function, R^{-1}.
(c) What are the domain and range of R^{-1}?
Figure 3.2: Graph of R(x)
x | 0 | 1 | 2 | 3
f(x) | 3 | 0 | 2 | -4
Table 3.14: f(x)
3. Refer to f in Table 3.14.
(a) What are the domain and range of f?
(b) Find a function table for the inverse function, f^{-1}.
(c) What are the domain and range of f^{-1}?

Explore the properties of functions with step-by-step solutions, including the domain, range, and inverse of linear and quadratic functions. Learn function composition and graphing techniques, including reflections for inverse functions. Suitable for high school students in Grades 10-12.

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