Graphing a Piecewise Function with Linear and Square Root Components

Math Problem Statement

Graph the piecewise function f(x) = { x - 8 for -1 < x < 2 ; sqrt(x - 3) + 4 for 3 ≤ x < 7 }.

The image presents a piecewise function $f(x)$ defined as:

\begin{cases} x - 8 & \text{for } -1 < x < 2 \\ \sqrt{x - 3} + 4 & \text{for } 3 \leq x < 7 \end{cases}$$ To graph this: ### 1. **Graph for $$-1 < x < 2$$ ( $$f(x) = x - 8$$ ):** - This is a linear equation with a slope of 1 and a y-intercept at $$-8$$. - For $$x = -1$$, $$f(x) = -1 - 8 = -9$$ (open point). - For $$x = 2$$, $$f(x) = 2 - 8 = -6$$ (open point). - You need to plot a line from $$(-1, -9)$$ to $$(2, -6)$$, but remember, both endpoints should be open because of the strict inequality. ### 2. **Graph for $$3 \leq x < 7$$ ( $$f(x) = \sqrt{x - 3} + 4$$ ):** - This is a square root function shifted right by 3 units and up by 4 units. - At $$x = 3$$, $$f(x) = \sqrt{3 - 3} + 4 = 4$$ (closed point). - At $$x = 7$$, $$f(x) = \sqrt{7 - 3} + 4 = \sqrt{4} + 4 = 6$$ (open point). - The curve should follow the shape of the square root function between $$x = 3$$ and $$x = 7$$. Would you like help with a specific part of the graph or need more details? ### 5 Related Questions: 1. How do you graph linear and square root functions individually? 2. What are open and closed points, and how do they affect a graph? 3. How does shifting affect the graph of a function? 4. Can you explain how to calculate the domain and range of piecewise functions? 5. How does the slope and y-intercept affect the appearance of a linear function? ### Tip: Always plot key points first and note whether they are open or closed when graphing piecewise functions.

Drop file here or Click Here to upload

Piecewise Functions
Linear Functions
Square Root Functions
Domain and Range

f(x) = x - 8 (for -1 < x < 2)
f(x) = sqrt(x - 3) + 4 (for 3 ≤ x < 7)

Properties of Linear Equations
Properties of Square Root Functions

Grades 9-12

Graph of a Piecewise Function with Linear, Square Root, and Quadratic Parts

Graphing Piecewise Function: f(x) = x+4 (x>1) and x-3 (x<=1)

Piecewise Function Graph: f(x) = { -3, x, 4 }

Graphing and Analyzing the Piecewise Function f(x) = x^3 - 1, 0, √(x + 1) - 2

Graphing a Piecewise Function with Quadratic and Linear Components