Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 35c

Chapter 3, Problem 35c

Find fg and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the composition of the functions f and g, denoted as fg(x) or f(g(x)). This means substituting g(x) into f(x).

Step 2: Write the expression for f(g(x)). Start with the definition of f(x), which is f(x) = 2x² − x − 3. Replace every occurrence of x in f(x) with g(x), which is x + 1. This gives f(g(x)) = 2(x + 1)² − (x + 1) − 3.

Step 3: Simplify the expression for f(g(x)). Begin by expanding (x + 1)² to get x² + 2x + 1. Substitute this back into the equation: f(g(x)) = 2(x² + 2x + 1) − (x + 1) − 3.

Step 4: Distribute and combine like terms. Expand 2(x² + 2x + 1) to get 2x² + 4x + 2. Then subtract (x + 1) and subtract 3. Combine all terms to simplify the expression for f(g(x)).

Step 5: Determine the domain of f(g(x)). The domain of g(x) is all real numbers since g(x) = x + 1 is a linear function. Similarly, the domain of f(x) is all real numbers since it is a quadratic function. Therefore, the domain of f(g(x)) is also all real numbers.

Verified video answer for a similar problem:

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

Video duration:

Was this helpful?

Key Concepts

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

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, fg means f(g(x)), which requires substituting g(x) into f(x). Understanding how to perform this substitution is crucial for finding the composed function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When composing functions, the domain of the resulting function fg must consider the domains of both f and g, ensuring that all inputs lead to valid outputs in the context of both functions.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The function f(x) = 2x² − x − 3 is a quadratic function, and its properties, such as its vertex and axis of symmetry, can influence the overall behavior of the composed function fg and its domain.

Recommended video:

06:36

Solving Quadratic Equations Using The Quadratic Formula

Related Practice

Textbook Question

Find f−g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

views

Textbook Question

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.

$f$\left$(x$\right$)=6x^2-x-1$ , $g$\left$(x$\right$)=x-1$

Textbook Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. x-intercept = -1/2 and y-intercept = 4

Textbook Question

Find ƒ+g, ƒ- g, ƒg and ƒ/g. Determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

views

Textbook Question

Find ƒ/g and determine the domain for each function. f(x) = 2x² − x − 3, g (x) = x + 1

Textbook Question

Write the standard form of the equation of the circle with the given center and radius. Center (-1, 4), r = 2

views