Math Problem Statement
Si dia la definizione di una funzione quadratica. Si dica inoltre se la funzione g(x) definita come g(x) = { x^2 + 2 se x < 2; 3x - 1 se x ≥ 2 } sia quadratica.
Solution
The piecewise function is defined as:
\begin{cases} x^2 + 2 & \text{se } x < 2 \\ 3x - 1 & \text{se } x \geq 2 \end{cases}$$ ### Key Observations: 1. **Quadratic Definition**: A quadratic function is any function that can be written in the form $$f(x) = ax^2 + bx + c$$, where $$a, b, c \in \mathbb{R}$$, and $$a \neq 0$$. - In this case, the portion $$x^2 + 2$$ (valid for $$x < 2$$) is a quadratic function because it has the term $$x^2$$ with $$a = 1$$. - The second part $$3x - 1$$ (valid for $$x \geq 2$$) is **not** quadratic; it is a linear function. 2. **Is $$g(x)$$ quadratic?** Since the definition of $$g(x)$$ includes both a quadratic expression and a linear expression, $$g(x)$$ **as a whole is not a quadratic function**. Quadratic functions must maintain the quadratic form for **all $$x$$** in their domain. --- ### Do you need further clarification or more examples? ### Related Questions: 1. How can we verify the continuity of $$g(x)$$ at $$x = 2$$? 2. What is the derivative of $$g(x)$$ for $$x < 2$$ and $$x \geq 2$$? 3. How do we determine if $$g(x)$$ is differentiable at $$x = 2$$? 4. Can you graph the piecewise function $$g(x)$$? 5. What is the value of $$g(x)$$ at specific points like $$x = 1$$ and $$x = 2$$? ### Tip: To determine if a function is continuous at a specific point $$x = a$$, check if the left-hand limit equals the right-hand limit and the function value at $$x = a$$.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Piecewise Functions
Continuity
Formulas
General form of a quadratic function: ax^2 + bx + c, where a ≠ 0
Theorems
Definition of quadratic and linear functions
Suitable Grade Level
Grades 9-11