Let's go over each part of the image and solve the problems.

Problem 5

The graph of $f(x)$ is provided, and we need to find the domain and range.

• Domain: The domain is the set of all possible input values (x-values) for which the function is defined. From the graph, the function is defined for $x$ values between $-5$ and $5$, inclusive.

  $\text{Domain: } [-5, 5]$

• Range: The range is the set of all possible output values (y-values) that the function can take. From the graph, the lowest point on the graph is at $y = -3$ and the highest point is at $y = 3$.

  $\text{Range: } [-3, 3]$

Problem 6

We are given the function $h(x) = -2f(x + 3) - 1$ and need to find the domain and range.

• Domain: The transformation $x + 3$ is a horizontal shift 3 units to the left. Therefore, the domain of $h(x)$ is the same as the domain of $f(x)$, shifted by 3 units to the left. The domain of $f(x)$ was $[-5, 5]$, so shifting this to the left by 3 gives:

  $\text{Domain: } [-8, 2]$

• Range: The transformation involves multiplying the function by $-2$, which reflects it over the x-axis and stretches it by a factor of 2. Additionally, subtracting 1 shifts the graph down by 1 unit. The range of $f(x)$ was $[-3, 3]$. After reflecting and stretching, this becomes $[-6, 6]$. Finally, subtracting 1 gives:

  $\text{Range: } [-7, -5]$

Problem 7

The function $k(x)$ is constructed by applying three transformations to $h(x)$:

• Horizontal dilation by a factor of 4,
• Vertical dilation by a factor of $\frac{1}{2}$,
• Vertical translation by 3 units.

The general form of $k(x)$ is given as $k(x) = ah(bx) + c$, and we need to find the values of $a$, $b$, and $c$.

• Horizontal dilation by 4 means $b = \frac{1}{4}$.
• Vertical dilation by $\frac{1}{2}$ means $a = \frac{1}{2}$.
• Vertical translation by 3 means $c = 3$.

Thus, the values are: $a = \frac{1}{2}, \quad b = \frac{1}{4}, \quad c = 3$

Would you like any further explanation or details?

Here are five related questions to deepen understanding:

1. How would the range change if $f(x)$ had a higher maximum point on the graph?
2. What happens to the domain if $h(x)$ had a transformation like $f(2x)$?
3. How does reflecting a function over the y-axis affect its domain and range?
4. Can you describe how to combine multiple transformations in a single step?
5. What is the effect of multiplying a function by a negative constant?

Tip: When working with transformations, it's helpful to apply one transformation at a time and see its effect on both the domain and range.