Math Problem Statement
Give the definiton of the image of a function f : [a, b] → R, and determine the image of the function f : [−1, 2] → R defined by f (x) = x2 − 4x + 1
Solution
Definition: Image of a Function
The image of a function is the set of all output values (range) that the function produces when applied to every input in the domain . Mathematically, the image of a function is written as:
Finding the Image of the Function
We are given the function and the domain . Let's determine its image:
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Rewrite the function: The function is a quadratic function. It can be rewritten in vertex form by completing the square: So, .
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Analyze the behavior on the domain :
- The vertex of the quadratic function is at , and the parabola opens upwards because the coefficient of is positive.
- At , the function attains its minimum value:
- Now, evaluate the function at the endpoints of the domain:
- At :
- At :
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Determine the range (image): Since the quadratic function is continuous and achieves its minimum value at , and the values at the endpoints are and , the image of the function over the interval is the interval:
Conclusion:
The image of the function on the domain is the interval .
Would you like more details on how the image is derived or have any questions?
Here are some related questions to consider:
- How do you complete the square for quadratic functions?
- How does the vertex form of a quadratic function help in determining its minimum or maximum?
- What are the properties of quadratic functions that determine the shape of their graphs?
- Can a quadratic function ever have an unbounded image on a closed interval?
- How do we compute the image of more complex functions?
Tip: When determining the image of a function, always check the behavior at critical points (like the vertex in a quadratic) and endpoints of the domain.
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Math Problem Analysis
Mathematical Concepts
Functions
Quadratic Functions
Image of a Function
Completing the Square
Formulas
f(x) = ax^2 + bx + c
Vertex form of a quadratic function: f(x) = (x - h)^2 + k
Image(f) = { f(x) | x ∈ [a, b] }
Theorems
Fundamental Theorem of Algebra (quadratic polynomials)
Properties of Quadratic Functions (vertex form, range)
Suitable Grade Level
Grades 10-12
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