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Starting with the graph of a basic function, graph the following function using the techniques of shifting, compressing, stretching, and/or reflecting. Find the domain and range of the function. \( f(x)=-(x+1)^{3}-4 \) Use the graphing tool to graph the equation. The domain of \( f(x) \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The range of \( f(x) \) is \( \square \). (Type your answer in interval notation.)

Ask by Cross Chen. in the United States
Feb 26,2025

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Answer

The function \( f(x) = -(x + 1)^3 - 4 \) is obtained by shifting the basic cubic function \( g(x) = x^3 \) one unit to the left, reflecting it across the x-axis, and then shifting it four units downward. The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \), and the range is also all real numbers, \( (-\infty, \infty) \).

Solution

We start with the basic cubic function, g(x) = x³, which has a domain of (–∞, ∞) and a range of (–∞, ∞). Now, consider the function:   f(x) = -(x + 1)³ – 4 Below are the step-by-step transformations: 1. Replace x by (x + 1) in x³.   • This shifts the graph 1 unit to the left.   • New graph: h(x) = (x + 1)³ 2. Multiply the function by –1.   • This reflects the graph across the x-axis.   • New graph: k(x) = –(x + 1)³ 3. Subtract 4 from the function.   • This shifts the graph 4 units downward.   • Final graph: f(x) = –(x + 1)³ – 4 Since cubic functions naturally extend in both vertical directions, these transformations do not restrict the function’s output. Therefore:  Domain of f(x): (–∞, ∞)  Range of f(x): (–∞, ∞) When using your graphing tool, you should see the basic cubic shape shifted, reflected, and translated according to the steps described.

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To transform the basic cubic function \( f(x) = x^3 \) into \( f(x) = -(x+1)^3 - 4 \), let's break it down step by step. 1. **Horizontal Shift**: The term \( (x + 1) \) indicates a left shift by 1 unit. 2. **Reflection**: The negative sign in front reflects the graph across the x-axis. 3. **Vertical Shift**: Lastly, subtracting 4 shifts the entire graph down by 4 units. So, starting with the graph of \( x^3 \), shift it left by 1, reflect it across the x-axis, and then shift it down by 4. The domain of \( f(x) \) is \( (-\infty, \infty) \)—cubic functions extend indefinitely in both horizontal directions. As for the range: since the highest point for \( -(x+1)^3 \) approaches \( +\infty \) and the lowest point can go down to \( -4 \), the range is \( (-\infty, -4] \). Range: \( (-\infty, -4] \)

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