College Algebra Study Guide: Functions, Polynomials, and Rational Functions

Functions and Their Inverses

Inverse Functions

An inverse function reverses the effect of the original function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Finding the Inverse: To find the inverse of $f(x)$, solve the equation $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

• Example 1: $f(x) = 5x + 8$ Solution: Set $y = 5x + 8$. Solve for $x$: $y = 5x + 8$ $y - 8 = 5x$ $x = \frac{y - 8}{5}$ Interchange $x$ and $y$: $f^{-1}(x) = \frac{x - 8}{5}$

• Example 2: $f(x) = \frac{3x+4}{2x-1}$ Solution: Set $y = \frac{3x+4}{2x-1}$ and solve for $x$: $y(2x-1) = 3x+4$ $2xy - y = 3x + 4$ $2xy - 3x = y + 4$ $x(2y - 3) = y + 4$ $x = \frac{y + 4}{2y - 3}$ Interchange $x$ and $y$: $f^{-1}(x) = \frac{x + 4}{2x - 3}$

Quadratic Functions and Parabolas

Vertex, Minimum/Maximum, Domain and Range

A quadratic function has the form $f(x) = ax^2 + bx + c$. Its graph is a parabola. The vertex is the highest or lowest point, depending on the sign of $a$.

• Vertex Formula: The vertex $(h, k)$ is given by: $h = -\frac{b}{2a}$ $k = f(h)$

• Minimum/Maximum: If $a > 0$, the parabola opens upward and has a minimum at the vertex. If $a < 0$, it opens downward and has a maximum at the vertex.

• Domain: For all quadratic functions, the domain is $(-\infty, \infty)$.

• Range: If the vertex is $(h, k)$ and $a > 0$, the range is $[k, \infty)$. If $a < 0$, the range is $(-\infty, k]$.

• Example 1: $f(x) = -3(x-2)^2 + 5$ This is in vertex form: $f(x) = a(x-h)^2 + k$ with $a = -3$, $h = 2$, $k = 5$. Vertex: $(2, 5)$ Maximum value: Since $a < 0$, maximum at vertex. Domain: $(-\infty, \infty)$ Range: $(-\infty, 5]$

• Example 2: $f(x) = -x^2 + 6x - 4$ $a = -1$, $b = 6$, $c = -4$ $h = -\frac{6}{2(-1)} = 3$ $k = f(3) = -(3)^2 + 6(3) - 4 = -9 + 18 - 4 = 5$ Vertex: $(3, 5)$ Maximum value: Since $a < 0$, maximum at vertex. Domain: $(-\infty, \infty)$ Range: $(-\infty, 5]$

Polynomial Functions

Key Characteristics of Polynomial Graphs

Polynomial functions are expressions of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$. Their graphs have distinct features based on degree and coefficients.

• Degree: The highest power of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• x-intercept(s): Values of $x$ where $f(x) = 0$.

• y-intercept: Value of $f(0)$.

• Example: For $f(x) = x^3 - 2x^2 + x - 5$:

  • Degree: 3

  • Leading Coefficient: 1

  • x-intercepts: Solve $x^3 - 2x^2 + x - 5 = 0$

  • y-intercept: $f(0) = -5$

Polynomial Division

Synthetic and Long Division

Polynomial division is used to divide one polynomial by another. Synthetic division is a shortcut for dividing by linear factors of the form $x - c$. Long division works for any divisor.

• Example: Divide $x^2 + 2x^2 - 5x - 6$ by $x + 1$. Long Division Steps:

  1. Divide the leading term of the dividend by the leading term of the divisor.

  2. Multiply the divisor by the result and subtract from the dividend.

  3. Repeat until the degree of the remainder is less than the degree of the divisor.

  Synthetic Division: Use $c = -1$ for $x + 1$.

Polynomial Construction

Factored Form from Zeros

A polynomial with real coefficients can be constructed from its zeros. If a polynomial has zeros at $x = a$, $x = b$, and $x = c$, then it can be written as $f(x) = k(x-a)(x-b)(x-c)$, where $k$ is a real constant.

• Example: Find a third-degree polynomial with zeros at $x = -3$, $x = 1$, and $x = 5$. $f(x) = k(x + 3)(x - 1)(x - 5)$ If $k = 1$, $f(x) = (x + 3)(x - 1)(x - 5)$

Rational Functions and Asymptotes

Vertical and Horizontal Asymptotes

Rational functions are quotients of polynomials. Their graphs may have vertical and horizontal asymptotes.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero. Example: For $f(x) = \frac{x+4}{x-2}$, vertical asymptote at $x = 2$.

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator.

  • If degree numerator < degree denominator: $y = 0$

  • If degrees are equal: $y = \frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}}$

  • If degree numerator > degree denominator: No horizontal asymptote

  Example: For $f(x) = \frac{2x^2+3}{x^2-1}$, degrees are equal, so horizontal asymptote at $y = \frac{2}{1} = 2$.

Graph Identification

Matching Functions to Graphs

To identify which graph matches a given function, analyze the degree, leading coefficient, and zeros. For $f(x) = x^3 - 4x^2$, the graph will:

• Be cubic (third degree)

• Have zeros at $x = 0$ and $x = 4$

• Leading coefficient is positive, so as $x \to \infty$, $f(x) \to \infty$

Compare these features to the provided graphs to select the correct one.