Let's analyze both functions to determine their domains and ranges.

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a) $f(x) = -3x + 8$

Domain:

The function $f(x) = -3x + 8$ is a linear function (a straight line). For any linear function, the domain is all real numbers because there are no restrictions on $x$.

• Domain: $(-\infty, \infty)$

Range:

Since the function is linear, it will keep increasing or decreasing indefinitely. For this specific function, the slope is negative ($-3$), meaning it decreases as $x$ increases. The output of the function can take any real value.

• Range: $(-\infty, \infty)$

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b) $f(x) = -0.5(x + 3)^2 + 4$

Domain:

The function $f(x) = -0.5(x + 3)^2 + 4$ is a quadratic function (a parabola). Quadratic functions are defined for all real numbers because there is no restriction on $x$.

• Domain: $(-\infty, \infty)$

Range:

The function is in the form of a parabola, and since the coefficient of $(x + 3)^2$ is negative ($-0.5$), it opens downward. The vertex will give us the maximum value of the function.

• The vertex occurs at $x = -3$, as that makes the square term zero.

• Substituting $x = -3$ into the function:

  $f(-3) = -0.5(-3 + 3)^2 + 4 = -0.5(0) + 4 = 4$

Since the parabola opens downward, the maximum value of $f(x)$ is 4. As $x$ moves away from $-3$, the value of $f(x)$ decreases indefinitely.

• Range: $(-\infty, 4]$

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Summary:

a) For $f(x) = -3x + 8$:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

b) For $f(x) = -0.5(x + 3)^2 + 4$:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, 4]$

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Would you like any further explanation or details on these functions? Here are a few related questions for deeper understanding:

1. How does the slope of a linear function affect its range?
2. What is the significance of the vertex in a quadratic function?
3. How do you determine the range of a quadratic function from its equation?
4. How does the direction in which a parabola opens affect the range?
5. How can you find the domain and range of rational functions?

Tip: For quadratic functions, the vertex gives key information about the maximum or minimum value of the function, depending on whether the parabola opens upward or downward.