Functions Cheatsheet

Cheatsheet Content

### Definition & Notation - **Definition:** A relation where each input (domain) has exactly one output (range). - **Notation:** $y = f(x)$, where $x$ is the independent variable and $y$ is the dependent variable. - **Domain:** Set of all possible input values for $x$. - **Range:** Set of all possible output values for $y$. ### Types of Functions #### Linear Functions - **Equation:** $y = mx + b$ - **Description:** Straight line. $m$ is slope, $b$ is y-intercept. - **Graph:** #### Quadratic Functions - **Equation:** $y = ax^2 + bx + c$ ($a \neq 0$) - **Description:** Parabola. Opens up if $a > 0$, down if $a #### Polynomial Functions - **Equation:** $P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ - **Description:** Degree $n$ (highest exponent). Smooth, continuous curves. - **End Behavior:** Determined by leading term $a_n x^n$. #### Rational Functions - **Equation:** $f(x) = P(x)/Q(x)$, where $P(x)$ and $Q(x)$ are polynomials, $Q(x) \neq 0$. - **Asymptotes:** - **Vertical:** $Q(x) = 0$ - **Horizontal/Slant:** Depends on degrees of $P(x)$ and $Q(x)$. - **Graph:** #### Exponential Functions - **Equation:** $y = a^x$ or $y = Ae^{kx}$ - **Description:** Growth ($a>1, k>0$) or decay ($0 #### Logarithmic Functions - **Equation:** $y = \log_b x$ (inverse of $b^y=x$) - **Description:** Inverse of exponential. - **Vertical Asymptote:** $x=0$ - **Graph:** #### Trigonometric Functions - **Basic:** $\sin(x), \cos(x), \tan(x)$ - **Periodic:** Repeat over an interval. - **Graph (Sine):** ### Function Operations - **Addition:** $(f+g)(x) = f(x) + g(x)$ - **Subtraction:** $(f-g)(x) = f(x) - g(x)$ - **Multiplication:** $(fg)(x) = f(x) \cdot g(x)$ - **Division:** $(f/g)(x) = f(x) / g(x)$, where $g(x) \neq 0$ - **Composition:** $(f \circ g)(x) = f(g(x))$ - Order matters! $(g \circ f)(x) = g(f(x))$ #### Example: If $f(x) = x^2$ and $g(x) = x+1$: - $(f+g)(x) = x^2 + x + 1$ - $(f \circ g)(x) = f(x+1) = (x+1)^2 = x^2 + 2x + 1$ - $(g \circ f)(x) = g(x^2) = x^2 + 1$ ### Transformations For a function $y = f(x)$: - **Vertical Shift:** $f(x) + c$ (up if $c>0$, down if $c 0$, left if $c 1$, compress if $0 1$, stretch if $0 ### Inverse Functions ($f^{-1}(x)$) - **Definition:** If $f(a)=b$, then $f^{-1}(b)=a$. - **Existence:** A function has an inverse if and only if it is one-to-one (passes horizontal line test). - **Finding Inverse:** 1. Replace $f(x)$ with $y$. 2. Swap $x$ and $y$. 3. Solve for $y$. 4. Replace $y$ with $f^{-1}(x)$. - **Graph:** Reflection across the line $y=x$. - **Example:** - $f(x) = 2x+3$ - $y = 2x+3$ - $x = 2y+3$ - $x-3 = 2y$ - $y = (x-3)/2$ - $f^{-1}(x) = (x-3)/2$ ### Piecewise Functions - **Definition:** Defined by multiple sub-functions, each applying to a different interval of the domain. - **Example:** $$ f(x) = \begin{cases} x^2 & \text{if } x ### Calculus Concepts #### Limits - **Definition:** The value a function approaches as the input approaches some value. - **Notation:** $\lim_{x \to c} f(x) = L$ - **Continuity:** A function is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$. #### Derivatives - **Definition:** Instantaneous rate of change, slope of tangent line. - **Notation:** $f'(x)$, $dy/dx$ - **Power Rule:** If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. - **Product Rule:** $(fg)' = f'g + fg'$ - **Quotient Rule:** $(f/g)' = (f'g - fg')/g^2$ - **Chain Rule:** $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ #### Integrals - **Definition:** Area under the curve, antiderivative. - **Notation:** $\int f(x) dx$ (indefinite), $\int_a^b f(x) dx$ (definite) - **Fundamental Theorem of Calculus:** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$.