Math Problem Statement
Graph the following function on the axes provided:
f(x) = { x + 11 for x < -5
-x + 8 for x ≥ -1 }
Solution
The piecewise function provided in your image is defined as:
\begin{cases} x + 11 & \text{for } x < -5 \\ -x + 8 & \text{for } x \geq -1 \end{cases}$$ We need to graph this on the provided axes. ### Step 1: Graph $$f(x) = x + 11$$ for $$x < -5$$ 1. The function $$f(x) = x + 11$$ is a linear function with a slope of 1 and a y-intercept of 11. 2. For $$x < -5$$, choose points to the left of $$x = -5$$ to plot: - When $$x = -6$$, $$f(-6) = -6 + 11 = 5$$. - When $$x = -7$$, $$f(-7) = -7 + 11 = 4$$. These points are on the line, so plot them and draw the line extending to the left of $$x = -5$$. 3. At $$x = -5$$, the function is not included (open circle), so we leave the endpoint open. ### Step 2: Graph $$f(x) = -x + 8$$ for $$x \geq -1$$ 1. The function $$f(x) = -x + 8$$ is also linear, with a slope of -1 and a y-intercept of 8. 2. For $$x \geq -1$$, choose points to the right of $$x = -1$$: - When $$x = -1$$, $$f(-1) = -(-1) + 8 = 9$$. - When $$x = 0$$, $$f(0) = -(0) + 8 = 8$$. - When $$x = 1$$, $$f(1) = -1 + 8 = 7$$. These points will form a line extending to the right of $$x = -1$$, and at $$x = -1$$, the endpoint is included (closed circle). --- Would you like any further clarification or additional details on graphing piecewise functions? Here are 5 questions to further your understanding: 1. How do you find the slope of a linear function? 2. What is the significance of open and closed circles in a piecewise graph? 3. How do we determine which points to choose when graphing linear functions? 4. What would happen if the function changed for values between $$-5$$ and $$-1$$? 5. How do you handle more complex functions, like quadratic or exponential, in a piecewise graph? **Tip:** Always check the boundary conditions carefully when graphing piecewise functions—whether the point is included or excluded makes a big difference!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Linear Functions
Inequalities in Functions
Formulas
Linear Function y = mx + b
Theorems
Properties of Piecewise Functions
Suitable Grade Level
Grades 9-10