Algebra of Functions, Composite Functions, and Inequalities: Study Guide

Section 3.5: Things to Know – Functions, Composite Functions, and Inequalities

Solving Compound Inequalities in One Variable

Compound inequalities involve two or more inequalities joined by "and" or "or". Solving these requires understanding how to combine solution sets.

• "And" Compound Inequality: Solution is the intersection of the individual solution sets.

• "Or" Compound Inequality: Solution is the union of the individual solution sets.

• Example: Solve $x > 2$ and $x < 5$; solution: $2 < x < 5$.

Solving Polynomial Inequalities

Polynomial inequalities require finding the intervals where the polynomial is positive or negative.

• Step 1: Set the inequality to zero: $x^3 - 3x^2 + 2x \\geq 0$.

• Step 2: Factor: $x(x-1)(x-2) \\geq 0$.

• Step 3: Find zeros: $x = 0, 1, 2$.

• Step 4: Test intervals between zeros to determine sign.

• Solution: $[0,1) \\cup [2, \\infty)$

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. The solution set excludes values that make the denominator zero.

• Example: Solve $\\frac{x+4}{x-1} \\leq 0$.

• Step 1: Find zeros and undefined points: $x+4=0$ ($x=-4$), $x-1=0$ ($x=1$).

• Step 2: Test intervals: $(-\\infty, -4)$, $(-4, 1)$, $(1, \\infty)$.

• Solution: $(-\\infty, -4] \\cup (1, \\infty)$

Determining the Domain and Range of a Function from Its Graph

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• To find the domain, look for all x-values covered by the graph.

• To find the range, look for all y-values covered by the graph.

• Example: If the graph extends from $x = -2$ to $x = 5$, domain is $[-2, 5]$.

Sketching the Graphs of Basic Functions

Basic functions include linear, quadratic, cubic, absolute value, square root, and reciprocal functions. Knowing their shapes helps in graphing and analyzing more complex functions.

• Linear: $f(x) = x$ (straight line)

• Quadratic: $f(x) = x^2$ (parabola)

• Cubic: $f(x) = x^3$ (S-shaped curve)

• Absolute Value: $f(x) = |x|$ (V-shaped)

• Square Root: $f(x) = \\sqrt{x}$ (half parabola)

• Reciprocal: $f(x) = \\frac{1}{x}$ (hyperbola)

Algebra of Functions

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \\cdot g(x)$

• Quotient: $\\left(\\frac{f}{g}\\right)(x) = \\frac{f(x)}{g(x)}$, $g(x) \\neq 0$

Example: If $f(x) = x^2$, $g(x) = 3x-1$:

• Sum: $x^2 + 3x - 1$

• Difference: $x^2 - 3x + 1$

• Product: $x^2(3x-1) = 3x^3 - x^2$

• Quotient: $\\frac{x^2}{3x-1}$

Evaluating Function Expressions

Given $f(x)$ and $g(x)$, you can evaluate expressions like $f+g$, $f-g$, $fg$, and $\\frac{f}{g}$ for specific values of $x$.

• Example: If $f(x) = \\frac{11}{x}$, $g(x) = \\sqrt{x}$, find:

• $a.$ $f+g(9) = f(9) + g(9) = \\frac{11}{9} + 3$

• $b.$ $f-g(1) = f(1) - g(1) = 11 - 1 = 10$

• $c.$ $\\frac{f}{g}(4) = \\frac{f(4)}{g(4)} = \\frac{11/4}{2} = \\frac{11}{8}$

• $d.$ $\\frac{g}{f}(0)$ is undefined since $f(0)$ is undefined.

Using Graphs to Evaluate Functions

To evaluate $f(x)$ or $g(x)$ at a specific $x$ using a graph:

• Find the $x$-value on the horizontal axis.

• Move vertically to the graph to find the corresponding $y$-value.

• If the graph does not exist at that $x$, the function is undefined there.

Intersections and Unions of Intervals

Interval Notation and Number Lines

Intervals are used to describe sets of numbers. The intersection ($\\cap$) of intervals is the set of numbers common to both.

• Example: $[0, \\infty) \\cap (-\\infty, 5] = [0,5]$

• Draw number lines to visualize intersections.

Set 1	Set 2	Intersection
[0, \\infty)	(-\\infty, 5]	[0,5]
((-\\infty, -2) \\cup (-2, \\infty))	(-4,0)	[-4,-2) \\cup (-2,0)

Domains of Combined and Composite Functions

Finding the Domain of Combined Functions

When combining functions, the domain of the result is the intersection of the domains of the individual functions, with additional restrictions from the operation.

• Sum/Difference: Domain is the intersection of the domains of $f$ and $g$.

• Product: Domain is the intersection of the domains of $f$ and $g$.

• Quotient: Domain is the intersection of the domains of $f$ and $g$, excluding values where $g(x) = 0$.

• Example: If $f(x) = \\sqrt{x+1}$ and $g(x) = x-16$:

• Domain of $f$: $x+1 \\geq 0 \\Rightarrow x \\geq -1$

• Domain of $g$: all real numbers

• Domain of $f+g$: $[-1, \\infty)$

• Domain of $\\frac{f}{g}$: $[-1, \\infty)$, $x \\neq 16$

Domains with Square Roots and Rational Expressions

For functions involving square roots, the radicand must be non-negative. For rational expressions, the denominator must not be zero.

• Example: $f(x) = \\sqrt{x-1}$, $g(x) = 4-x$

• Domain of $f$: $x-1 \\geq 0 \\Rightarrow x \\geq 1$

• Domain of $g$: all real numbers

• Domain of $f+g$: $[1, \\infty)$

• Domain of $f-g$: $[1, \\infty)$

• Domain of $\\frac{f}{g}$: $[1,4)$ (since $g(x)=0$ at $x=4$)

Composite Functions and Their Domains

A composite function $f \\circ g$ is defined as $f(g(x))$. The domain of $f \\circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

• Example: $f(x) = \\sqrt{x}$, $g(x) = 4-x$

• $f \\circ g(x) = f(g(x)) = \\sqrt{4-x}$

• Domain: $4-x \\geq 0 \\Rightarrow x \\leq 4$

• Domain: $(-\\infty, 4]$

Summary Table: Domains of Combined Functions

Operation	Domain Rule	Example
Sum/Difference	Intersection of domains	$f(x) = \\sqrt{x-1}$, $g(x) = 4-x$; domain: $[1, \\infty)$
Product	Intersection of domains	Same as above
Quotient	Intersection, exclude $g(x)=0$	Domain: $[1,4)$
Composite	$x$ in domain of $g$, $g(x)$ in domain of $f$	Domain: $(-\\infty, 4]$

Additional info: These notes cover essential College Algebra topics including inequalities, function operations, domains, and interval notation, with worked examples and graphical interpretations.