Introduction to Functions and Graphs

Functions and their graphical representations form the foundation of calculus. Understanding the properties and behavior of functions is crucial for solving problems and analyzing real-world situations. This study guide will provide an overview of the key concepts and techniques related to functions and graphs in Calculus I.

Common Terms and Definitions

Function: A rule that assigns a unique output value to each input value in its domain.

Domain: The set of all possible input values for a function.

Range: The set of all possible output values for a function.

Independent Variable: The input variable of a function, typically denoted by x.

Dependent Variable: The output variable of a function, typically denoted by y or f(x).

Continuous Function: A function whose graph can be drawn without lifting the pen from the paper.

Discontinuous Function: A function whose graph has breaks, jumps, or gaps.

Types of Functions

Linear Function: A function of the form f(x) = mx + b, where m is the slope and b is the y-intercept.

Quadratic Function: A function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0.

Polynomial Function: A function consisting of terms with non-negative integer exponents, such as f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0.

Rational Function: A function that can be expressed as the ratio of two polynomial functions, such as f(x) = (ax + b) / (cx + d).

Exponential Function: A function of the form f(x) = a^x, where a is a positive constant.

Logarithmic Function: A function of the form f(x) = log_a(x), where a is a positive constant, and a ≠ 1.

Trigonometric Function: Functions that relate angles to the lengths of sides in a right triangle, such as sine, cosine, and tangent.

Graphing Techniques

1. Identify the domain and range of the function.
2. Find the x- and y-intercepts by setting y = 0 and x = 0, respectively.
3. Determine the symmetry of the function (even, odd, or neither).
4. Identify asymptotes (vertical, horizontal, or oblique) by examining the function's behavior as x approaches infinity or specific values.
5. Calculate the function's values for key points and plot them on the coordinate plane.
6. Connect the points smoothly, taking into account the function's properties and behavior.

Common Questions and Answers

What is the difference between a function and an equation?

A function is a rule that assigns a unique output value to each input value in its domain, while an equation is a statement that two mathematical expressions are equal.

How can I determine if a function is continuous or discontinuous?

A function is continuous if its graph can be drawn without lifting the pen from the paper. If there are breaks, jumps, or gaps in the graph, the function is discontinuous. You can also check for continuity by examining the function's domain and ensuring that the limit of the function exists at every point in the domain.

What are the steps to graph a function?

To graph a function, first identify its domain and range. Then, find the x- and y-intercepts, determine the function's symmetry, and identify any asymptotes. Calculate the function's values for key points and plot them on the coordinate plane. Finally, connect the points smoothly, taking into account the function's properties and behavior.

Conclusion

Functions and their graphical representations are essential concepts in Calculus I. By understanding the different types of functions, their properties, and graphing techniques, you will be well-prepared to tackle a wide range of problems in calculus and its applications. Practice graphing various functions and analyzing their behavior to reinforce your understanding of these fundamental concepts.