https://file.aiquickdraw.com/mathbot/file/1726275632944crdwrmi6.jpg

For part 6a, we are provided with the graph of \( f(x) \), and the domain and range are already noted. Let's review that first before moving on to parts 6b and 6c:

### Part 6a Review:
- **Domain**: The domain of a function is the set of all possible input values (x-values). Based on the graph, the function \( f(x) \) is defined from \( x = -6 \) to \( x = 3 \), so the domain is:
  \[
  \text{Domain: } \{ x \mid -6 \leq x \leq 3 \}
  \]
- **Range**: The range of a function is the set of all possible output values (y-values). From the graph, the minimum y-value is \( -2 \) and the function extends upwards indefinitely (as there's no cap on y in the positive direction). Therefore, the range is:
  \[
  \text{Range: } \{ y \mid y \geq -2 \}
  \]

### Part 6b: \( -f(x) + 3 \)
This transformation involves two operations:
1. \( -f(x) \) reflects the graph of \( f(x) \) across the x-axis.
2. Adding 3 to the result shifts the entire graph upwards by 3 units.

#### Domain of \( -f(x) + 3 \):
Since reflecting and shifting the graph doesn't affect the x-values, the domain remains the same:
\[
\text{Domain: } \{ x \mid -6 \leq x \leq 3 \}
\]

#### Range of \( -f(x) + 3 \):
The range is affected by both the reflection and the vertical shift. The original range of \( f(x) \) was \( y \geq -2 \). Reflecting it gives a range of \( y \leq 2 \), and then shifting it up by 3 units changes the range to:
\[
\text{Range: } \{ y \mid y \leq 5 \}
\]

### Part 6c: \( f(-x) - 2 \)
This transformation involves two operations:
1. \( f(-x) \) reflects the graph of \( f(x) \) across the y-axis.
2. Subtracting 2 shifts the entire graph downward by 2 units.

#### Domain of \( f(-x) - 2 \):
The domain reflects across the y-axis. Originally, the domain of \( f(x) \) was \( -6 \leq x \leq 3 \). Reflecting it gives a domain of \( -3 \leq x \leq 6 \), so the domain becomes:
\[
\text{Domain: } \{ x \mid -3 \leq x \leq 6 \}
\]

#### Range of \( f(-x) - 2 \):
The range was \( y \geq -2 \). Shifting the entire graph downward by 2 units gives a new range:
\[
\text{Range: } \{ y \mid y \geq -4 \}
\]

Would you like further explanation or to work through the next part?

Here are 5 related questions to explore:
1. How does reflecting across the x-axis affect the range of a function?
2. How do vertical shifts impact both the domain and range of a function?
3. What changes occur in the domain when reflecting across the y-axis?
4. How can you identify transformations such as reflections and shifts on a graph?
5. What are the similarities and differences between horizontal and vertical transformations?

**Tip**: To understand transformations better, try sketching the graph of each transformation to visualize how it affects the original function.

Explore the domain and range for the functions -f(x) + 3 and f(-x) - 2 in a high school-level math problem. Learn about function transformations, including reflections and vertical shifts, with detailed explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation