Key Concepts in College Algebra: Rational, Polynomial, Exponential, and Logarithmic Functions

Rational Functions

Graphing Rational Functions

Rational functions are functions of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. Their graphs can exhibit vertical and horizontal asymptotes, and may have holes or intercepts depending on the numerator and denominator.

• Vertical Asymptotes: Occur at values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Example: For $f(x) = \frac{7x^2 - 7x - 210}{x^2 + 6x - 7}$, set the denominator equal to zero to find vertical asymptotes: $x^2 + 6x - 7 = 0 \implies x = 1, x = -7$.

• Horizontal Asymptote: Since degrees of numerator and denominator are equal, divide leading coefficients: $y = \frac{7}{1} = 7$.

Additional info: Holes occur where both numerator and denominator share a common factor.

Polynomial Functions

Intermediate Value Theorem and Real Zeros

The Intermediate Value Theorem states that if a function $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then there exists at least one $c$ in $(a, b)$ such that $f(c) = 0$.

• Application: To determine if $f(x) = 5x^4 - 9x^2 - 19$ has a real zero between $x = -2$ and $x = 1$, evaluate $f(-2)$ and $f(1)$.

• If $f(-2)$ and $f(1)$ have opposite signs, a real zero exists in $(-2, 1)$.

Additional info: This theorem is fundamental for locating roots of continuous functions.

Exponential Functions

Transformations and Asymptotes

Exponential functions have the form $f(x) = a^{x}$, where $a > 0$ and $a \neq 1$. Transformations can shift the graph vertically or horizontally, affecting the domain, range, and asymptotes.

• Vertical Shifts: $g(x) = e^{x} + k$ shifts the graph up by $k$ units.

• Horizontal Asymptote: For $g(x) = e^{x} + 15$, the horizontal asymptote is $y = 15$.

• Domain: $(-\infty, \infty)$ for all exponential functions.

• Range: $(15, \infty)$ for $g(x) = e^{x} + 15$.

• Example: $g(x) = e^{x} - 15$ has horizontal asymptote $y = -15$ and range $(-15, \infty)$.

Logarithmic Functions

Graphing and Transformations

Logarithmic functions are the inverses of exponential functions and have the form $f(x) = \log_b(x)$, where $b > 0$, $b \neq 1$. Transformations affect the position and shape of the graph.

• Vertical Shifts: $f(x) = \log_{1/4}(x + 5) - 4$ shifts the graph down by 4 units.

• Horizontal Shifts: The graph shifts left by 5 units due to $(x + 5)$.

• Vertical Asymptote: Occurs at $x = -5$.

• Domain: $x > -5$.

• Range: $(-\infty, \infty)$ for all logarithmic functions.

Additional info: The base $1/4$ indicates a decreasing logarithmic function.

Solving Equations Using Logarithms

Isolating Variables in Exponential Equations

Logarithms are useful for solving equations where the variable is in the exponent. The change of base formula and properties of logarithms allow for isolating the variable.

• Example: Given $M = R \left( \frac{a}{b} + 1 \right)^{x}$, solve for $x$:

$M = R \left( \frac{a}{b} + 1 \right)^{x}$ Divide both sides by $R$: $\frac{M}{R} = \left( \frac{a}{b} + 1 \right)^{x}$ Take logarithms of both sides: $x = \frac{\log(M/R)}{\log(a/b + 1)}$

• Key Point: Use logarithms with appropriate bases to solve for exponents.

Evaluating Functions with Logarithmic Inputs

Substitution and Simplification

To evaluate a function at a logarithmic input, substitute the value and simplify using properties of logarithms and exponents.

• Example: For $f(x) = 4^{x}$, find $f(\log_{4} 3 \ln 4)$.

Substitute $x = \log_{4} 3 \ln 4$: $f(x) = 4^{\log_{4} 3 \ln 4}$ Recall $a^{\log_{a} b} = b$ and $\ln 4$ is a constant multiplier. $4^{\log_{4} 3 \ln 4} = (4^{\log_{4} 3})^{\ln 4} = 3^{\ln 4}$ $3^{\ln 4} = e^{\ln 4 \cdot \ln 3}$

• Key Point: Use exponent and logarithm properties to simplify complex function inputs.