Math Problem Statement

Formulate the graph of f(x) = { (x+1, if x<-5), ((x-1)^2 + 1, if -5 <= x < 5), (x+1, if x >= 5) }.

Solution

The piecewise function given is:

x + 1, & \text{if } x < -5 \\ (x - 1)^2 + 1, & \text{if } -5 \leq x < 5 \\ x + 1, & \text{if } x \geq 5 \end{cases}$$ ### Let's break this down: 1. **For $$x < -5$$**: - The function is linear: $$f(x) = x + 1$$. - This represents a line with slope 1 and y-intercept at $$y = 1$$. 2. **For $$-5 \leq x < 5$$**: - The function is quadratic: $$f(x) = (x - 1)^2 + 1$$. - This represents a parabola shifted 1 unit to the right and 1 unit up. The vertex of the parabola is at $$(1, 1)$$. 3. **For $$x \geq 5$$**: - The function is linear again: $$f(x) = x + 1$$, just like in the first interval. - This is the same linear line as for $$x < -5$$. ### Key Points to Graph: - **First region** $$x < -5$$: Graph the line $$f(x) = x + 1$$. - **Middle region** $$-5 \leq x < 5$$: Graph the parabola $$f(x) = (x - 1)^2 + 1$$. - **Last region** $$x \geq 5$$: Graph the same line $$f(x) = x + 1$$. ### Discontinuities: - There could be potential discontinuities at $$x = -5$$ and $$x = 5$$. Let’s check the values: - For $$x = -5$$: - From the first part, $$f(x) = (-5 + 1) = -4$$. - From the quadratic part, $$f(-5) = ((-5 - 1)^2) + 1 = 36 + 1 = 37$$. - So, there is a **jump discontinuity** at $$x = -5$$. - For $$x = 5$$: - From the quadratic part, $$f(5) = (5 - 1)^2 + 1 = 16 + 1 = 17$$. - From the third part, $$f(x) = 5 + 1 = 6$$. - There is also a **jump discontinuity** at $$x = 5$$. Would you like me to provide a detailed plot of this function, or explain any part further? ### Questions for further exploration: 1. How do we determine whether a function has a discontinuity at a given point? 2. What does the vertex of a parabola represent, and how is it affected by shifts? 3. How would you find the slope of the lines for $$x < -5$$ and $$x \geq 5$$? 4. How does the quadratic part of the function differ from the linear parts? 5. Can a function have multiple types of discontinuities? ### Tip: When graphing piecewise functions, always check for discontinuities by calculating the left-hand and right-hand limits at the transition points.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Quadratic Functions
Discontinuities

Formulas

f(x) = x + 1 for x < -5 and x >= 5
f(x) = (x - 1)^2 + 1 for -5 ≤ x < 5

Theorems

Piecewise Function Analysis
Quadratic Vertex Formula
Continuity and Discontinuity at Transition Points

Suitable Grade Level

Grades 10-12

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