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: $(-\infty, \infty)$

(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: $[0, \infty)$

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.