Math Problem Statement
Sketching graphs of piecewise functions. For example, the given function is:
y =
{ x - 1, if x < 1;
3x, if x >= 1 }
Determine whether the piecewise function is continuous or not.
Solution
The question in the image involves sketching and analyzing a piecewise function and determining its continuity. Let's break it down step by step.
The given function is:
\begin{cases} x - 1, & \text{if } x < 1, \\ 3x, & \text{if } x \geq 1. \end{cases}$$ ### (a) **Sketching the graph of the piecewise function** 1. **First Piece**: $$y = x - 1$$ for $$x < 1$$: - This is a linear equation with a slope of 1 and a y-intercept of $$-1$$. - For $$x < 1$$, the function ends (but does not include) the point $$(1, 0)$$. 2. **Second Piece**: $$y = 3x$$ for $$x \geq 1$$: - This is also a linear equation with a slope of 3 and a y-intercept of 0. - For $$x \geq 1$$, the function starts at $$(1, 3)$$ and continues. **Key Transition Point:** At $$x = 1$$, we check where the two parts connect: - From the first piece, $$y = x - 1$$, so $$y = 0$$ at $$x = 1$$. - From the second piece, $$y = 3x$$, so $$y = 3$$ at $$x = 1$$. Therefore, the graph has a **discontinuity** at $$x = 1$$ since the two parts do not meet. ### (b) **Determining continuity** A piecewise function is **continuous** if there are no gaps or jumps at the points where the pieces meet. For $$x = 1$$: - **Left-hand limit**: $$\lim_{x \to 1^-} (x - 1) = 0$$, - **Right-hand limit**: $$\lim_{x \to 1^+} (3x) = 3$$, - Since the left-hand and right-hand limits are not equal, the function is **not continuous** at $$x = 1$$. ### Final Notes: - The graph will show a jump at $$x = 1$$, with a hole at $$(1, 0)$$ and a solid point at $$(1, 3)$$. --- Would you like a detailed sketch of the graph or further explanations? --- ### 5 Related Questions: 1. How do you formally define continuity for a piecewise function at a specific point? 2. How can you check if a piecewise function is differentiable? 3. What is the effect of changing the slopes of the lines in the given piecewise function? 4. How do we find the domain and range of this function? 5. Can a function be discontinuous but still have a defined limit at the discontinuity? ### Tip: When analyzing a piecewise function, always check the transition points by comparing the left-hand limit, right-hand limit, and the actual value of the function at those points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Graph sketching
Continuity of functions
Formulas
y = x - 1 for x < 1
y = 3x for x ≥ 1
Theorems
Definition of continuity
Limits and continuity
Suitable Grade Level
Grades 9-12
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