Let's determine the domain and range for each function given in the image.

1. $f(x) = -|x - 2| + 3$

• Domain: The absolute value function is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$
• Range: The absolute value function produces only non-negative values. Since it is negated ($-|x - 2|$), it reflects downward. The maximum value occurs at $x = 2$, where $f(2) = 3$. As $x \to \pm\infty$, $f(x) \to -\infty$. $\text{Range: } (-\infty, 3]$

2. $f(x) = 4x - 3$

• Domain: This is a linear function, which is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$
• Range: A linear function can take all real values. $\text{Range: } (-\infty, \infty)$

3. $y = 4x^2 - 2$

• Domain: A quadratic function is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$
• Range: The quadratic function $y = 4x^2 - 2$ is an upward-facing parabola with a minimum at $x = 0$. At $x = 0$, $y = -2$, and as $|x| \to \infty$, $y \to \infty$. $\text{Range: } [-2, \infty)$

Would you like a deeper explanation for any of these? Let me know!

Related Questions:

1. How do you determine the domain of a function with a fraction?
2. What transformations affect the range of an absolute value function?
3. How does the coefficient of $x^2$ in a quadratic function affect its range?
4. What is the general approach to finding the range of a function?
5. How do you determine if a function is increasing or decreasing?

Math Tip:

For absolute value functions of the form $f(x) = a |x - h| + k$, the range is affected by $k$ (vertical shift) and $a$ (reflection and stretch/compression).