Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 38b

Chapter 3, Problem 38b

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3| b. f(-5)

Verified step by step guidance

Step 1: Understand the function f(x) = |x + 3| / |x + 3|. This function involves absolute values in both the numerator and denominator. The absolute value of a number is its distance from zero, so it is always non-negative.

Step 2: Substitute the given value of x = -5 into the function. This means replacing x with -5 in both the numerator and denominator: f(-5) = |(-5) + 3| / |(-5) + 3|.

Step 3: Simplify the expressions inside the absolute value bars. For the numerator and denominator, calculate (-5) + 3, which equals -2. So the function becomes f(-5) = |-2| / |-2|.

Step 4: Evaluate the absolute values. The absolute value of -2 is 2, so the function simplifies to f(-5) = 2 / 2.

Step 5: Simplify the fraction. Divide the numerator by the denominator to find the simplified value of the function at x = -5.

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Key Concepts

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

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. For example, |3| = 3 and |-3| = 3. This function is crucial in evaluating expressions that involve both positive and negative values, as it affects the outcome based on the sign of the input.

Evaluating Functions

Evaluating a function involves substituting a specific value for the independent variable (in this case, x) into the function's expression. This process allows us to determine the corresponding output value. For instance, to evaluate f(-5), we replace x with -5 in the function f(x) = |x + 3| / |x + 3| and simplify the expression accordingly.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, often by performing arithmetic operations and applying algebraic rules. In the context of the given function, simplification may include canceling common factors or reducing fractions. Understanding how to simplify is essential for accurately determining the value of the function after evaluation.

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Textbook Question

In Exercises 27–38, evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/|x + 3| a. f(5)

Textbook Question

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Textbook Question

Evaluate each function at the given values of the independent variable and simplify. f(x) = |x+3|/(x + 3) c. f(−9 - x)

Textbook Question

Graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

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Textbook Question

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

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Textbook Question

Write the standard form of the equation of the circle with the given center and radius. Center (-4, 0), r = 10

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