Math Problem Statement

Use the same scales and axes to draw the graphs of the two functions f(x) = 2 + 2x + x^2 and g(x) = 5 - 2x - x^2 in the domain -3 ≤ x ≤ 2, x ∈ R. (a) Use your graphs to estimate the values of x for which f(x) = g(x). (b) Use your graphs to estimate the values of x for which f(x) ≥ g(x).

Solution

The intersection points, where f(x)=g(x)f(x) = g(x), are approximately:

  1. x=1+1020.42x = -1 + \frac{\sqrt{10}}{2} \approx -0.42
  2. x=11022.58x = -1 - \frac{\sqrt{10}}{2} \approx -2.58

Answers for each part:

  • (a) The values of xx for which f(x)=g(x)f(x) = g(x) are approximately x0.42x \approx -0.42 and x2.58x \approx -2.58.
  • (b) For f(x)g(x)f(x) \geq g(x), the intervals can be estimated from the graph. It appears that f(x)g(x)f(x) \geq g(x) when x[3,2.58][0.42,2]x \in [-3, -2.58] \cup [-0.42, 2].

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Math Problem Analysis

Mathematical Concepts

Algebra
Graphing Functions
Quadratic Equations

Formulas

Quadratic functions f(x) = ax^2 + bx + c

Theorems

Intersection of functions

Suitable Grade Level

Grades 9-12