Increasing and Decreasing Functions

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Questions and Answers

What is the shape of the graph of the function f(x) = x^2, x ∈ R?

  • Parabola (correct)
  • Circle
  • Hyperbola
  • Line

What happens to the height of the graph as we move from left to right along the graph to the right of the origin?

  • It first increases and then decreases
  • It increases (correct)
  • It remains constant
  • It decreases

What can be said about the function f(x) = x^2, x ∈ R, for x < 0?

  • It is decreasing (correct)
  • It is constant
  • It is strictly decreasing
  • It is increasing

What is the analytical definition of a function that is increasing on an interval I?

<p>x1 &lt; x2 in I ⇒ f(x1) &lt; f(x2) for all x1, x2 ∈ I (D)</p> Signup and view all the answers

What is the analytical definition of a function that is decreasing on an interval I?

<p>x1 &lt; x2 in I ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ I (B)</p> Signup and view all the answers

What happens to the height of the graph as we move from left to right along the graph to the left of the origin?

<p>It decreases (B)</p> Signup and view all the answers

What is the analytical definition of a function that is constant on an interval I?

<p>f(x) = c for all x ∈ I, where c is a constant (A)</p> Signup and view all the answers

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Study Notes

Analyzing Functions Using Differentiation

  • The function f(x) = x^2, x ∈ R is a parabola.

Graph Analysis

  • To the right of the origin, the graph continuously increases as we move from left to right.
  • To the left of the origin, the graph continuously decreases as we move from left to right.

Definitions of Increasing, Decreasing, and Constant Functions

Increasing Function

  • A function f is increasing on an interval I if x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I.

Decreasing Function

  • A function f is decreasing on an interval I if x1 < x2 in I ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ I.
  • A function f is strictly decreasing on an interval I if x1 < x2 in I ⇒ f(x1) > f(x2) for all x1, x2 ∈ I.

Constant Function

  • A function f is constant on an interval I if f(x) = c for all x ∈ I, where c is a constant.

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