College Algebra: Unit 1 Exam Review Study Notes

Section 1.2: Functions and Graphs

Evaluating Functions

Evaluating a function involves substituting a given value for the independent variable and simplifying the expression.

• Key Point: For $f(x) = x^2 - x$, to evaluate at $x = 1$, substitute and simplify: $f(1) = 1^2 - 1 = 0$.

• Example: If $f(x) = x^2 - x$, then $f(1) = 0$.

Interpreting Graphs

Graphs can be used to analyze the behavior of functions and answer questions about real-world scenarios.

• Key Point: The graph of student enrollment percentage over time can be used to estimate values and trends.

• Example: If the graph shows 22% in 1997, then $f(1997) = 22$.

Section 1.3: Domain, Range, and Difference Quotient

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Key Point: Use the graph to determine the domain and range. For example, if the graph covers $x$ from $0$to $4$, then the domain is $[0,4]$.

• Example: If the graph passes the vertical line test, it represents a function.

Difference Quotient

The difference quotient is a fundamental concept in calculus and algebra, used to find the average rate of change of a function.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2 + x$, the difference quotient is $\frac{(x+h)^2 + (x+h) - (x^2 + x)}{h}$.

Section 1.4: Linear Equations and Graphing

Point-Slope Form

The point-slope form of a line is useful for writing the equation of a line given a point and a slope.

• Formula: $y - y_1 = m(x - x_1)$

• Example: Passing through $(3,5)$ and $(6,8)$, first find the slope $m = \frac{8-5}{6-3} = 1$, then use the formula.

Graphing Linear Equations

To graph a linear equation, plot points and draw the line through them.

• Key Point: For $y = \frac{1}{2}x + 1$, plot the y-intercept at $(0,1)$ and use the slope to find another point.

Modeling with Linear Functions

Linear functions can model real-world scenarios, such as depreciation or value over time.

• Example: If a car's value decreases linearly from $14,760$ in 1995 to $4440$ in 1999, find the slope and write the equation.

Section 1.5: Slope-Intercept Form and Average Rate of Change

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: For a line perpendicular to $y = 2x + 8$, the slope is $-\frac{1}{2}$ (negative reciprocal).

Average Rate of Change

The average rate of change of a function $f(x)$ from $x=a$ to $x=b$ is given by:

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = \sqrt{x}$ from $x=2$ to $x=8$, $\frac{\sqrt{8} - \sqrt{2}}{8-2}$

Tabular Data: Charitable Contributions

This table shows charitable contributions over several years, useful for calculating average rates of increase.

Year	Charitable Contributions
1991	$2480
1992	$2540
1993	$2480
1994	$2580
1995	$2630
1996	$3130

Main Purpose: To compare yearly contributions and calculate average rate of increase.

Section 1.7: Function Composition

Composition of Functions

Function composition involves applying one function to the result of another function.

• Formula: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = x^2 + 2x - 2$ and $g(x) = x^2 - 2x - 4$, then $f(g(x-2))$ means substitute $x-2$ into $g$, then use that result in $f$.

Expressing Functions as Compositions

Some functions can be written as the composition of two or more functions.

• Example: If $h(x) = \sqrt{2x+5}$, then $h(x) = f(g(x))$ where $g(x) = 2x+5$ and $f(x) = \sqrt{x}$.

Section 1.8: Inverse Functions

Finding Inverse Functions

An inverse function reverses the effect of the original function. To find the inverse, solve for $x$ in terms of $y$ and interchange variables.

• Example: For $f(x) = (x+1)^2$, solve $y = (x+1)^2$ for $x$ to get $f^{-1}(y) = \sqrt{y} - 1$.

Additional info:

• All sections and questions are directly relevant to College Algebra topics, including functions, graphing, linear equations, composition, and inverses.

• Tabular data was interpreted for average rate of change calculations.