Graphing a Piecewise Function with Quadratic and Linear Components

Math Problem Statement

4. Gambarkan grafik fungsi f(x) = { x^2 , x < 1; 2 , x = 1; -x + 1 , x > 1 }

The problem asks us to graph the piecewise function $f(x)$ defined as follows:

\begin{cases} x^2 & , \, x < 1 \\ 2 & , \, x = 1 \\ -x + 1 & , \, x > 1 \end{cases}$$ To create the graph of $$f(x)$$, let's analyze each part of this piecewise function: 1. **For $$x < 1$$**: The function $$f(x) = x^2$$ represents a parabola. We only consider this part of the parabola for values of $$x$$ less than 1. As $$x$$ approaches 1 from the left, $$f(x) \to 1$$. 2. **For $$x = 1$$**: The function has the value $$f(x) = 2$$. This is a single point at $$(1, 2)$$, which we’ll plot as a dot on the graph. 3. **For $$x > 1$$**: The function $$f(x) = -x + 1$$ is a line with a slope of -1 and a y-intercept of 1. This line only applies for $$x > 1$$. ### Graphing Strategy - Plot the curve $$f(x) = x^2$$ for $$x < 1$$. - Mark the point $$(1, 2)$$ as an isolated point on the graph. - Draw the line $$f(x) = -x + 1$$ for $$x > 1$$. Would you like me to create this graph for you, or do you have any specific questions about the steps? --- ### Related Questions 1. How would the graph change if $$f(x) = x^2$$ applied for $$x \leq 1$$? 2. What happens to the function if the point at $$x = 1$$ is removed? 3. How do we determine the continuity of this piecewise function? 4. How does the slope of $$-x + 1$$ affect the graph for $$x > 1$$? 5. What other piecewise functions involve quadratic and linear components? ### Tip When graphing piecewise functions, make sure to mark any isolated points separately from continuous parts.

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Piecewise Functions
Graphing Functions
Quadratic Functions
Linear Functions

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

Piecewise Function Analysis

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