Functions and Graphs

Find the domain of y = (x + 3) / (4 − √(x² − 9)).

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Identify the expression in the denominator: 4 - √(x² - 9). The denominator cannot be zero, so set up the inequality 4 - √(x² - 9) ≠ 0.

Solve the inequality 4 - √(x² - 9) ≠ 0 by first isolating the square root: √(x² - 9) ≠ 4.

Square both sides of the inequality to eliminate the square root: x² - 9 ≠ 16.

Solve the resulting inequality: x² ≠ 25. This implies x ≠ 5 and x ≠ -5.

Consider the domain of the square root function itself: x² - 9 must be greater than or equal to 0, which means x ≥ 3 or x ≤ -3. Combine this with the previous result to find the domain of the function.

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Key Concepts

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In this case, we need to determine the values of x that do not lead to undefined expressions, such as division by zero or taking the square root of a negative number.

Square Root Function

The square root function, denoted as √(x), is defined only for non-negative values of x. This means that for the expression √(x² - 9) to be valid, the quantity inside the square root must be greater than or equal to zero, leading to the inequality x² - 9 ≥ 0.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function y = (x + 3) / (4 - √(x² - 9)) is rational, and we must ensure that the denominator (4 - √(x² - 9)) does not equal zero, as this would make the function undefined.

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Textbook Question

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A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.

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Textbook Question

Piecewise-Defined Functions

In Exercises 35 and 36, find the (a) domain and (b) range.

𝔂 = { √ -x, -4 ≤ x ≤ 0

{ √ x, 0 < x ≤ 4

Textbook Question

Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?

a. ƒg

b. ƒ³

c. ƒ(sin x)

d. g(sec x)