Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Combining Functions

Calculus

0. Functions

Combining Functions

Previous problemNext problem

2:43 minutes

Problem 1.2.79b

Textbook Question

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

b. f/g

Verified step by step guidance

First, recall the definitions of even and odd functions. An even function satisfies f(x) = f(-x) for all x, while an odd function satisfies g(x) = -g(-x) for all x.

Consider the function h(x) = f(x)/g(x). To determine if h(x) is even or odd, we need to evaluate h(-x) and compare it to h(x).

Calculate h(-x): h(-x) = f(-x)/g(-x). Since f is even, f(-x) = f(x). Since g is odd, g(-x) = -g(x). Therefore, h(-x) = f(x)/(-g(x)).

Simplify h(-x): h(-x) = -f(x)/g(x). Notice that h(-x) = -h(x), which is the definition of an odd function.

Conclude that the function h(x) = f(x)/g(x) is odd, given that f is even and g is odd.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Even and Odd Functions

An even function satisfies the property f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. An odd function satisfies g(-x) = -g(x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for analyzing the behavior of combined functions.

Function Division

When dividing two functions, such as f/g, the resulting function is defined only where g(x) is not zero. The properties of the original functions (even or odd) can influence the nature of the quotient, but one must also consider the points where the denominator may affect the overall function's parity.

Parity of Combined Functions

The parity of combined functions can be determined by analyzing their definitions. For instance, the quotient f/g will be even if both f and g are even, or if f is even and g is odd. Conversely, it will be odd if f is odd and g is even. This requires careful consideration of the definitions and properties of the functions involved.

Watch next

Master Adding & Subtracting Functions with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Composition of Functions

Copy and complete the following table.

a. <IMAGE>

Textbook Question

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

c. g(g(−1))

Textbook Question

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

g. g ∘ f

Textbook Question

Combining Functions

Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?

d. f² = ff

Textbook Question

Composition of Functions

A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.

Textbook Question

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

f. f(g(1/2))

Textbook Question

Composition of Functions

Evaluate each expression using the functions

f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0

x − 1, 0 ≤ x ≤ 2

e. g(f(0))