Math Problem Statement

Find the range of the function for f(x) = 10 - x^2 and g(x) = 5 + sqrt(4 - x).

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. f(x)=10x2f(x) = 10 - x^2

This is a downward-opening parabola. The range depends on the vertex and the direction of the parabola.

  • The function is of the form f(x)=a(xh)2+kf(x) = a(x-h)^2 + k, where a=1a = -1, h=0h = 0, and k=10k = 10.
  • The vertex is at (0,10)(0, 10), and since a=1a = -1, the parabola opens downward.

The maximum value of f(x)f(x) is at the vertex, where f(0)=10f(0) = 10. As x2x^2 grows larger, f(x)f(x) decreases without bound.

Range of f(x)f(x): (,10](-\infty, 10]


18. g(x)=5+4xg(x) = 5 + \sqrt{4 - x}

For this function, the square root limits the domain since the expression under the square root, 4x4 - x, must be non-negative.

  • The condition for the square root is 4x04 - x \geq 0, or x4x \leq 4.
  • The minimum value of g(x)g(x) occurs when x=4x = 4, giving g(4)=5+0=5g(4) = 5 + 0 = 5.
  • The maximum value occurs when x=x = -\infty, though it approaches a limiting value.

Range of g(x)g(x): [5,)[5, \infty)


Would you like to explore more details about these ranges or move on to the remaining exercises? Here are some related questions:

  1. What is the vertex form of a quadratic equation?
  2. How do we determine the domain of a square root function?
  3. What transformations affect the range of a quadratic function?
  4. How does the range of a function relate to its graph?
  5. 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