Functions and Graphs: College Algebra Chapter 1 Review

Functions and Graphs

Evaluating Functions

Evaluating a function involves substituting a given value for the variable and calculating the result. Functions are often written as f(x), where x is the input variable.

• Definition: A function f assigns each input x exactly one output f(x).

• Example: For $f(x) = x^2 - 4x - 7$, to find $f(-3)$, substitute $x = -3$:

$f(-3) = (-3)^2 - 4(-3) - 7 = 9 + 12 - 7 = 14$

• Difference Quotient: The difference quotient is used to find the average rate of change of a function:

$rac{f(x+h) - f(x)}{h}$

Graphical Analysis of Functions

Graphs provide visual representations of functions, allowing analysis of domain, range, intercepts, and symmetry.

• Domain: The set of all possible input values (x) for which the function is defined.

• Range: The set of all possible output values (f(x)).

• X-intercepts: Points where the graph crosses the x-axis ($f(x) = 0$).

• Y-intercepts: Points where the graph crosses the y-axis ($x = 0$).

• Example: If a graph passes through (0, 4), the y-intercept is 4.

Symmetry and Function Classification

Functions can be classified as even, odd, or neither based on their symmetry.

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric about the origin.

• Neither: If neither condition is met.

• Example: $f(x) = x^2 - 4x - 7$ is neither even nor odd.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Example: $f(x) = egin{cases} x + 6 & ext{if } x < -2 \ x - 6 & ext{if } x ext{≥} -2 \\ ext{(Additional info: Piecewise functions are graphed by plotting each piece over its respective interval.)} \\ ext{Range is determined by the output values for all intervals.} \\ ext{Domain is the union of all intervals.} \\

Linear Functions and Slope-Intercept Form

Linear functions are written in the form $y = mx + b$, where m is the slope and b is the y-intercept.

• Slope: Measures the steepness of the line ($m = rac{ ext{rise}}{ ext{run}}$).

• Y-intercept: The value of $y$ when $x = 0$.

• Example: For $y = rac{1}{2}x + 5$, slope is $rac{1}{2}$, y-intercept is 5.

Writing Equations of Lines

Equations of lines can be written given a point and a slope, or two points.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals.

• Example: A line parallel to $y = 3x - 7$ passing through (2, 3) is $y = 3x - 3$.

Average Rate of Change

The average rate of change of a function between $x = a$ and $x = b$ is:

$rac{f(b) - f(a)}{b - a}$

Domain and Interval Notation

Domain and range are often expressed in interval notation.

• Example: For $f(x) = rac{4}{x-7}$, the domain is all real numbers except $x = 7$.

• Interval Notation: $(a, b)$ for open intervals, $[a, b]$ for closed intervals.

Function Operations and Composition

Functions can be added, subtracted, multiplied, divided, and composed.

• Sum: $(f+g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $( rac{f}{g})(x) = rac{f(x)}{g(x)}$

• Composition: $(f igcirc g)(x) = f(g(x))$

Inverse Functions

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(y)$ maps $y$ back to $x$.

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then replace $y$ with $x$.

• Example: If $f(x) = x^2 - 5$, then $f^{-1}(x) = ext{(Additional info: Only for$x ext{≥} 0$)} \\ f^{-1}(x) = ext{(solving$x = y^2 - 5$for$y$)} \\ y^2 = x + 5 \\ y = ext{(Additional info:$y = ext{sqrt}(x+5)$for$y ext{≥} 0$)}$

Summary Table: Function Properties

Property	Definition	Example
Domain	Set of all input values	$(-, 8)$ for $f(x) = x^2$
Range	Set of all output values	$[0, 8)$ for $f(x) = x^2$
X-intercept	Where $f(x) = 0$	$(0, 0)$ for $f(x) = x$
Y-intercept	Where $x = 0$	$(0, b)$ for $y = mx + b$
Even Function	$f(-x) = f(x)$	$f(x) = x^2$
Odd Function	$f(-x) = -f(x)$	$f(x) = x^3$
Piecewise	Defined by intervals	$f(x) = egin{cases} x+6 & x<-2 \ x-6 & x ext{≥}-2 \\ ext{(Additional info: See above)} \\

Additional info:

• Some examples and explanations have been expanded for clarity and completeness.

• Graphical analysis includes identifying intervals of increase/decrease, relative minima/maxima, and constant intervals.

• Inverse functions require the original function to be one-to-one (pass the horizontal line test).