Precalculus Benchmark Skills: Comprehensive Study Guide

Functions and Their Properties

Evaluating Functions

Functions assign each input exactly one output. Evaluating a function means finding the output for a given input value.

• Definition: A function f(x) gives a unique value for each x in its domain.

• Example: If $f(x) = 2x + 3$, then $f(4) = 2(4) + 3 = 11$.

Piecewise Functions

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

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \geq 0 \end{cases}$

• To evaluate, determine which piece applies to the input value.

Translations of Functions

Translations shift the graph of a function horizontally or vertically.

• Quadratic Function: $f(x) = x^2$

• Translation: $f(x) = (x - h)^2 + k$ shifts right by h and up by k.

• Square Root Function: $f(x) = \sqrt{x}$; $f(x) = \sqrt{x - h} + k$ shifts right by h and up by k.

Algebra of Functions and Domains

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the combined function is the intersection of the domains of the original functions.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Product: $(fg)(x) = f(x)g(x)$

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

Inverse Functions

Two functions are inverses if their composition yields the identity function: $f(g(x)) = g(f(x)) = x$.

• Test: Substitute one function into the other and simplify.

Polynomial and Rational Functions

Polynomial Division and Asymptotes

Long division is used to divide polynomials. Vertical asymptotes occur where the denominator of a rational function is zero (and the numerator is nonzero).

• Example: Divide $x^3 + 2x^2 - x + 1$ by $x - 1$ using long division.

• Vertical Asymptote: For $f(x) = \frac{1}{x-2}$, the vertical asymptote is $x = 2$.

Polynomial and Rational Inequalities

To solve inequalities, find the zeros of the numerator and denominator, then test intervals.

• Example: Solve $\frac{x-1}{x+2} > 0$.

Exponential and Logarithmic Functions

Solving Exponential Equations

Exponential equations can be solved by expressing both sides with the same base or by using logarithms.

• With Logarithms: $2^x = 7 \implies x = \log_2 7$

• Without Logarithms: $2^x = 8 \implies x = 3$ (since $8 = 2^3$)

Solving Logarithmic Equations

Use properties of logarithms to combine or expand terms, then exponentiate both sides to solve for the variable.

• Example: $\log(x) + \log(2) = 1 \implies \log(2x) = 1 \implies 2x = 10 \implies x = 5$

Matrices and Systems of Equations

Matrix Operations

Matrices can be added, subtracted, and multiplied (if dimensions are compatible).

• Addition: Add corresponding elements.

• Multiplication: Multiply rows by columns.

Determinant of a Matrix

The determinant is a scalar value that can be computed from a square matrix and is used in solving systems of equations.

• 2x2 Matrix: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

• 3x3 Matrix (Expansion by Minors): $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$

Solving Matrix Equations

Matrix equations of the form $AX = B$ can be solved using the inverse of A, if it exists: $X = A^{-1}B$.

Cramer's Rule

Cramer's Rule uses determinants to solve systems of linear equations.

• For two variables: $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, where D is the determinant of the coefficient matrix.

Partial Fraction Decomposition

Express a rational function as a sum of simpler fractions.

• Example: $\frac{3x+5}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2}$

Nonlinear Systems and Graphing Inequalities

Nonlinear systems may involve equations like circles and lines. Systems of inequalities are graphed by shading the solution regions.

• Solving by Addition: Add or subtract equations to eliminate a variable.

• Graphing: Shade the region that satisfies all inequalities.

Topic	Key Concept	Example
Function Evaluation	Substitute input into function	$f(2) = 2^2 + 1 = 5$
Matrix Determinant	Calculate for square matrix	$\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = 1*4 - 2*3 = -2$
Exponential Equation	Use logarithms to solve	$3^x = 20 \implies x = \log_3 20$
Partial Fractions	Decompose rational expression	$\frac{5}{x^2-1} = \frac{A}{x-1} + \frac{B}{x+1}$