Math Problem Statement
Solution
The exercises in the image ask to find the range of the given functions for problems 17 to 20. Let's solve each function step by step:
17.
This is a downward-opening parabola. The range depends on the vertex and the direction of the parabola.
- The function is of the form , where , , and .
- The vertex is at , and since , the parabola opens downward.
The maximum value of is at the vertex, where . As grows larger, decreases without bound.
Range of :
18.
For this function, the square root limits the domain since the expression under the square root, , must be non-negative.
- The condition for the square root is , or .
- The minimum value of occurs when , giving .
- The maximum value occurs when , though it approaches a limiting value.
Range of :
Would you like to explore more details about these ranges or move on to the remaining exercises? Here are some related questions:
- What is the vertex form of a quadratic equation?
- How do we determine the domain of a square root function?
- What transformations affect the range of a quadratic function?
- How does the range of a function relate to its graph?
- How can we restrict the domain of a function to change its range?
Tip: Always check the domain of a function first, especially if it involves square roots or fractions.
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Math Problem Analysis
Mathematical Concepts
Algebra
Functions
Quadratic Functions
Square Root Functions
Formulas
f(x) = a(x-h)^2 + k for quadratic functions
g(x) = sqrt(4 - x)
Theorems
Properties of quadratic functions
Square root domain theorem
Suitable Grade Level
Grades 9-11