Which of the following is the graph of the fuctions f(x) defined below?
`f(x) = {:(x^(2) - 2, "for", x le 1),(x-7, "for",1 < x < 5),(4 - 7 ,"for", x ge 5):}`

Options

A.
B.
C.
D.

Answer

To determine the correct graph of the piecewise function \( f(x) \) defined as:

\[ f(x) = \begin{cases} x^2 - 2 & \text{for } x \leq 1 \\ x - 7 & \text{for } 1 < x < 5 \\ 4 - x & \text{for } x \geq 5 \end{cases} \]

we will analyze each piece of the function step by step.

### Step 1: Analyze the first piece \( f(x) = x^2 - 2 \) for \( x \leq 1 \)

1. **Determine the shape**: The function \( x^2 - 2 \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive).
2. **Find the vertex**: The vertex occurs at \( x = 0 \), where \( f(0) = 0^2 - 2 = -2 \).
3. **Calculate the endpoint**: At \( x = 1 \), \( f(1) = 1^2 - 2 = -1 \).
4. **Plot the points**: The graph will include the point \( (1, -1) \) and will extend to the left, approaching \( (-\infty, -2) \).

### Step 2: Analyze the second piece \( f(x) = x - 7 \) for \( 1 < x < 5 \)

1. **Determine the shape**: This is a linear function with a slope of 1.
2. **Calculate the endpoints**:
- At \( x = 1 \): \( f(1) = 1 - 7 = -6 \) (not included, as the interval is open).
- At \( x = 5 \): \( f(5) = 5 - 7 = -2 \) (not included, as the interval is open).
3. **Plot the points**: The graph will be a line segment starting just above \( (1, -6) \) and ending just below \( (5, -2) \).

### Step 3: Analyze the third piece \( f(x) = 4 - x \) for \( x \geq 5 \)

1. **Determine the shape**: This is also a linear function with a negative slope of -1.
2. **Calculate the endpoint**: At \( x = 5 \), \( f(5) = 4 - 5 = -1 \) (this point is included).
3. **Plot the points**: The graph will start at \( (5, -1) \) and extend downward as \( x \) increases.

### Step 4: Combine the pieces

Now we combine the three pieces:

- For \( x \leq 1 \): The parabola \( y = x^2 - 2 \) from \( (-\infty, -2) \) to \( (1, -1) \).
- For \( 1 < x < 5 \): The line segment from just above \( (1, -6) \) to just below \( (5, -2) \).
- For \( x \geq 5 \): The line starting from \( (5, -1) \) and extending downward.

### Step 5: Identify the correct graph

From the analysis, we can conclude:

- The graph must show a parabola that opens upwards for \( x \leq 1 \).
- The segment for \( 1 < x < 5 \) must be a line that does not include the endpoints.
- The graph for \( x \geq 5 \) must start at \( (5, -1) \) and continue downward.

### Conclusion

After analyzing all three pieces of the function, we can determine which graph corresponds to the function \( f(x) \). The correct graph will be the one that matches the described characteristics.