Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 24

Chapter 3, Problem 24

For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. ƒ(x)=√(5x-4), g(x)=-(1/x)

Verified step by step guidance

First, write down the given functions clearly: $f(x) = \sqrt{5x - 4}$ and $g(x) = -\frac{1}{x}$.

To find $(f+g)(x)$, add the two functions: $(f+g)(x) = \sqrt{5x - 4} - \frac{1}{x}$.

To find $(f-g)(x)$, subtract $g(x)$ from $f(x)$: $(f-g)(x) = \sqrt{5x - 4} + \frac{1}{x}$ (note the minus sign in $g(x)$).

To find $(fg)(x)$, multiply the two functions: $(fg)(x) = \sqrt{5x - 4} \times \left(-\frac{1}{x}\right) = -\frac{\sqrt{5x - 4}}{x}$.

To find $(f/g)(x)$, divide $f(x)$ by $g(x)$: $(f/g)(x) = \frac{\sqrt{5x - 4}}{-\frac{1}{x}} = -x \sqrt{5x - 4}$. Then, determine the domain for each by considering the restrictions: for $f(x)$, the expression inside the square root must be non-negative ($5x - 4 \geq 0$), and for $g(x)$, $x \neq 0$ because of the denominator. Also, for $(f/g)(x)$, ensure $g(x) \neq 0$.

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Key Concepts

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

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division to create new functions. For example, (ƒ+g)(x) = ƒ(x) + g(x) and (f/g)(x) = ƒ(x) divided by g(x), provided g(x) ≠ 0. Understanding these operations is essential to manipulate and analyze combined functions.

Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. When combining functions, the domain of the resulting function is the intersection of the individual domains, considering restrictions like division by zero or square roots of negative numbers.

Square Root and Rational Function Restrictions

Square root functions require the radicand to be non-negative, so for ƒ(x) = √(5x-4), 5x-4 ≥ 0. Rational functions like g(x) = -1/x are undefined when the denominator is zero, so x ≠ 0. These restrictions must be considered when determining the domain of combined functions.

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Related Practice

Textbook Question

Determine whether the three points are the vertices of a right triangle. (-4,1),(1,4),(-6,-1)

Textbook Question

Determine whether each relation defines a function, and give the domain and range.

Textbook Question

Determine whether the three points are the vertices of a right triangle. See Example 3.

$(- 2, - 8), (0, - 4), (- 4, - 7)$

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Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x²

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Textbook Question

Graph each piecewise-defined function.

$f (x) = {\begin{matrix} 2 x + 1 & if x \geq 0 \\ x & if x < 0 \end{matrix}$

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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. horizontal, through (-7,4)