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

1:48 minutes

Problem 41

Textbook Question

Composite functions and notation
Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
ƒ (√(x+4))

Verified step by step guidance

Identify the function \( f(x) = x^2 - 4 \).

Recognize that you need to evaluate \( f(\sqrt{x+4}) \).

Substitute \( \sqrt{x+4} \) into the function \( f(x) \), replacing \( x \) with \( \sqrt{x+4} \).

The expression becomes \( (\sqrt{x+4})^2 - 4 \).

Simplify the expression by calculating \( (\sqrt{x+4})^2 \) which results in \( x+4 \), and then subtract 4.

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.

Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, if we have functions f and g, the composite function f(g(x)) means we first evaluate g at x and then apply f to that result. Understanding how to manipulate and simplify composite functions is crucial for evaluating expressions like f(√(x+4)).

Function Notation

Function notation is a way to denote functions and their operations clearly. For example, f(x) represents the function f evaluated at x. This notation allows us to express complex operations succinctly, such as f(√(x+4)), which indicates that we need to substitute √(x+4) into the function f. Mastery of function notation is essential for working with multiple functions.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with or evaluate. This can include combining like terms, factoring, or substituting values. In the context of the given problem, simplifying f(√(x+4)) requires substituting √(x+4) into the function f and then performing algebraic operations to arrive at a more manageable expression.

Watch next

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

Start learning

Related Videos

Related Practice

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(y⁴)

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(g(y))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

g(ƒ(u))

Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

F(F(x))

Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (x³ - 5)¹⁰

Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

Textbook Question

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g .

h(x) = √ (x⁴ + 2 )

Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o g

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

Composite functions and notationLet ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).Simplify or evaluate the following expressions.ƒ (√(x+4))

Key Concepts

Composite Functions

Function Notation

Simplification of Expressions

Watch next

Composite functions and notation
Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
ƒ (√(x+4))