Precalculus Practice Quizzes

Domain and Range of a Function

- The domain of a function is the set of all values for which the function is defined.
- The range of the function is the set of all values that the function takes.
- This quiz has questions with domains and ranges of polynomial functions, functions with absolute value and functions with radicals.
Domain and range of a function practice

Inequalities Problems

- Linear inequalities.
- Parabolic inequalities.
- Inequalities with absolute values.
Inequalities Problems

Polynomial and Rational Inequalities Problems

To solve a polynomial or rational inequality,
1. Factor the polynomial completely over the real numbers.
2. Mark the zeros of the polynomial on a number line.
3. Determine the sign of the polynomial on each of the resulting intervals.
4. Select the intervals corresponding to the sign of the original inequality.
Polynomial and Rational Inequalities

Inverse Functions Problems

- A function f(x) is one-to-one if it does not assign the same value to two different elements of its domain.
- If f(x) is a 1-1 function, then it has an inverse function f^-1 defined by f^-1(y)=x if f(x)=y, for all y in the range of f.
- The domain of f^-1 is the range of f, and the range of f^-1 is the domain of f.
- To find a formula for f^-1, we can set y = f(x), solve for x in terms of y, and set f^-1(y) = x.
Inverse Functions Quiz

Algebraic Simplification Quiz

In the following quiz, "algebraic simplification" means you should:
- eliminate compound fractions,
- factor as much as possible,
- put terms over a common denominator when feasible,
- transform negative exponents.
Algebraic Simplification Quiz

Quadratic Functions Practice

- find the coordinates of the vertex of a quadratic function,
- find the minimum or the maximum of a quadratic function,
- find a function given the coordinates of the vertex and of a point on the parabola, or the coordinates of the vertex and of the minimum/maximum,
- find the points of intersection of parabolas with lines or with other parabolas.
Quadratic Functions Practice

Translation and Reflection of Graphs Practice

Translations of graphs:
Translate a graph m (m > 0) units to the right, by replacing x with x - m.
Translate a graph m (m > 0) units to the left, by replacing x with x + m.
Translate a graph m (m > 0) units up, by replacing y with y - m.
Translate a graph m (m > 0) units down, by replacing y with y + m.

Reflections of graphs:
Reflect a graph in the y-axis by replacing x with -x.
Reflect a graph in the x-axis by replacing y with -y.
Reflect a graph in the x = y line by replacing x with y and y with x.
Translation and Reflection of Graphs Practice

Roots and Radicals Equations Practice

In most cases, a square root in an equation can be eliminated by squaring. However, this method can sometimes produce solutions that don't work named extraneous roots, so all solutions need to be verified.
If the equation has two or more roots, the squaring process can be repeated for each of them.
Roots and Radicals Equations Practice

Properties of Logarithms and Exponents Practice

Properties of logarithms and exponents:

Properties of Logarithms and Exponents Practice

Logarithmic and Exponential Equations Practice

Equations involving logarithms and exponentials:
Logarithmic and Exponential Equations Practice

Trigonometric Functions of Angles Practice

- The sine function is positive in Quadrants I and II, the cosine function is positive in Quadrants I and IV, and the tangent function is positive in Quadrants I and III.
- The following quiz asks students to calculate the cosinus, sinus, tangent, cotangent, secant and cosecant of angles that are multiples of 30^o, 45^o, 60^o and 90^o.
Trigonometric Functions of Angles Practice

Trigonometric Addition Formulas Practice

Addition and subtraction formulas for sine, cosine and tangent.

Trigonometric Addition Formulas Practice

Trigonometric Equations Practice

Trigonometric equations can be solved by using standard algebraic techniques such as collecting like terms and factoring. The initial goal in solving a trigonometric equation is to isolate the trigonometric function in the equation.

Example:

Because cos(x) has a period of 2π, first find all solutions in the interval [0, 2π).
The solutions are π/4 and 7π/4.

The general solution of the equation can be found by adding multiples of 2π to each of these solutions: π/4 + 2nπ and 7π/4 + 2nπ.

Trigonometric Equations Practice