https://file.aiquickdraw.com/mathbot/file/1726988402538saqntqiv.jpg

Let's analyze and solve the problem step-by-step.

### Problem (from the image):
You are asked to sketch the graph and find the domain and range of each of the following functions:
1. \( f(x) = 2x - 1 \)
2. \( f(x) = x^2 \)

### (a) \( f(x) = 2x - 1 \)
#### 1. Graph:
- This is a **linear function**, where the slope is \( 2 \) and the y-intercept is \( -1 \).
- The line passes through the points:
  - \( x = 0 \): \( f(0) = -1 \) (so the line crosses the y-axis at \( (0, -1) \)).
  - \( x = 1 \): \( f(1) = 2(1) - 1 = 1 \) (so the line passes through \( (1, 1) \)).

The graph is a straight line with slope 2, slanting upwards from left to right.

#### 2. Domain:
- The domain of any linear function is all real numbers.
- **Domain**: \( (-\infty, \infty) \)

#### 3. Range:
- Since the function is linear and continuous without any restrictions, the range also includes all real numbers.
- **Range**: \( (-\infty, \infty) \)

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### (b) \( f(x) = x^2 \)
#### 1. Graph:
- This is a **parabola** that opens upwards.
- It has a vertex at \( (0, 0) \), and it is symmetric about the y-axis.
- The parabola passes through:
  - \( (1, 1) \) because \( f(1) = 1^2 = 1 \).
  - \( (-1, 1) \) because \( f(-1) = (-1)^2 = 1 \).
  - As \( |x| \) increases, the function values \( f(x) = x^2 \) also increase.

The graph of this function is a U-shaped curve opening upwards.

#### 2. Domain:
- The function \( f(x) = x^2 \) is defined for all real values of \( x \).
- **Domain**: \( (-\infty, \infty) \)

#### 3. Range:
- Since the function is a parabola that opens upwards, the lowest value of \( f(x) \) is 0 (at \( x = 0 \)), and it increases as \( x \) moves away from 0.
- **Range**: \( [0, \infty) \)

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### Summary:

- **(a) \( f(x) = 2x - 1 \)**:
  - Domain: \( (-\infty, \infty) \)
  - Range: \( (-\infty, \infty) \)

- **(b) \( f(x) = x^2 \)**:
  - Domain: \( (-\infty, \infty) \)
  - Range: \( [0, \infty) \)

Would you like me to provide a detailed graph for both functions?

Here are 5 related questions to expand your understanding:
1. How does the slope of a linear function affect the steepness of the graph?
2. What happens to the graph of \( f(x) = x^2 \) if you add or subtract a constant (e.g., \( f(x) = x^2 + 1 \))?
3. How would the graph of \( f(x) = 2x - 1 \) change if the slope is negative?
4. Can you find the x-intercept of \( f(x) = 2x - 1 \) and explain its significance?
5. How do you determine the vertex of a quadratic function like \( f(x) = x^2 + 4x + 3 \)?

### Tip:
When analyzing the domain and range, always check for restrictions on the inputs (e.g., square roots or denominators) to make sure the function is defined for all values.

Sketch the graph and find the domain and range of each function: (a) f(x) = 2x - 1, (b) f(x) = x^2

Learn how to graph the linear function f(x) = 2x - 1 and the quadratic function f(x) = x^2. This guide explains how to find the domain and range of these functions with step-by-step instructions, suitable for students in Grades 8-10.

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