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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 16
Chapter 3, Problem 16

Let ƒ(x)=x2+3ƒ(x)=x^2+3 and g(x)=−2x+6g(x)=-2x+6. Find each of the following. See Example 1.
(ƒg)(−3) (ƒg)(-3)

Verified step by step guidance
1
Understand that (ƒg)(x) means the composition of functions ƒ and g, which is ƒ(g(x)). This means you first apply g to x, then apply ƒ to the result.
Start by finding g(-3) by substituting -3 into the function g(x) = -2x + 6. Write the expression: \(g(-3) = -2(-3) + 6\).
Simplify the expression for g(-3) to find the value that will be input into ƒ. This involves multiplying and adding the numbers inside the parentheses.
Next, substitute the value found for g(-3) into the function ƒ(x) = x^2 + 3. Write the expression: \(ƒ(g(-3)) = (g(-3))^2 + 3\).
Finally, simplify the expression by squaring the value of g(-3) and then adding 3 to find the value of (ƒg)(-3).

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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 (ƒg)(x) = ƒ(g(x)). It means you first evaluate g at x, then use that output as the input for ƒ. This concept is essential for understanding how to combine functions and evaluate them at specific values.
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Evaluating Functions at a Given Input

Evaluating a function at a given input means substituting the input value into the function's formula and simplifying to find the output. For example, to find g(-3), replace x with -3 in g(x) = -2x + 6 and simplify. This step is crucial before composing functions.
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Polynomial and Linear Functions

ƒ(x) = x² + 3 is a polynomial function, and g(x) = -2x + 6 is a linear function. Understanding their forms helps in correctly substituting and simplifying expressions during evaluation and composition. Recognizing these types aids in anticipating the behavior and complexity of the functions.
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Related Practice
Textbook Question

Determine the intervals of the domain over which each function is continuous.

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Textbook Question

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. P(-5,-6), Q(7,-1)

Textbook Question

Determine the intervals of the domain over which each function is continuous.

Textbook Question

Determine whether each relation defines a function. {(-12,5),(-10,3),(8,3)}

Textbook Question

Let ƒ(x)=x2+3 and g(x)=-2x+6. Find each of the following. (ƒ/g)(-1)

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Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. through (-8,4), undefined slope

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