Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 5

Chapter 3, Problem 5

Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$
$open paren f with hook composed with g close paren times 2$

Verified step by step guidance

Understand that the notation $(\f\circ g)(2)$ means the composition of the functions $f$ and $g$ evaluated at $x=2$. This is written as $f(g(2))$, which means you first find $g(2)$ and then plug that result into $f$.

Start by evaluating $g(2)$. Since $g(x) = x^2$, substitute $2$ for $x$ to get $g(2) = 2^2$.

Calculate \$2^2$ to find the value of $g(2)$. This gives you the input for the next step.

Next, substitute the value of $g(2)$ into the function $f(x) = x + 1$. So, you will compute $f(g(2)) = f(\text{value from previous step}) = \text{value} + 1$.

Finally, add 1 to the value obtained from $g(2)$ to complete the evaluation of $(f \circ g)(2)$.

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Key Concepts

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (f∘g)(x) = f(g(x)). It means you first evaluate g at x, then use that output as the input for f. Understanding this process is essential to correctly evaluate composite functions.

Evaluating Functions

Evaluating a function means substituting the input value into the function's formula and simplifying. For example, if f(x) = x + 1, then f(3) = 3 + 1 = 4. This skill is necessary to find the value of functions at specific points.

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed, typically parentheses, exponents, multiplication/division, and addition/subtraction. Correctly applying this order ensures accurate evaluation of expressions like g(2) = 2^2 before applying f.

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Related Practice

Textbook Question

Fill in the blank to correctly complete each sentence. The y-intercept of the graph of y = -2x + 6 is ________.

Textbook Question

Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$

$open paren f with hook slash g close paren times 2$

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. For the function ƒ(x) = -4x + 2, ƒ(-2)= ______.

Textbook Question

Fill in the blank to correctly complete each sentence. The x-intercept of the graph of 2x + 5y = 10 is ________.

Textbook Question

Fill in the blank(s) to correctly complete each sentence. The equation y = 4x - 6 defines a function with independent variable______ and dependent variable ________ .

Textbook Question

Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$

$open paren g composed with f with hook close paren times 2$

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Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1ƒ(x)=x+1 and g(x)=x2g(x)=x^2 (ƒ∘g)(2)(ƒ∘g)(2)

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Key Concepts

Function Composition

Evaluating Functions

Order of Operations

Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$
$open paren f with hook composed with g close paren times 2$