LT 2.1/2.2

2.1 Functions

Function Evaluation and Piecewise Functions

• Evaluating Functions: Substitute the specified input value into every occurrence of the independent variable and simplify using standard order of operations.

  • Example: Given $f(x) = 2x^2 - 3x + 1$ , evaluate $f(4)$.     $f(4) = 2(4)^2 - 3(4) + 1 = 32 - 12 + 1 = 21$

• Piecewise Functions: Functions defined by different formulas or rules over separate intervals of their domain.

  • To evaluate, determine which sub-domain interval contains the input $x$, then evaluate using only that corresponding rule.

  • Example:     $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 - 3 & x \ge 0 \end{cases}$

  • For $x = -2$: since $-2 < 0$, evaluate $f(-2) = 2(-2) + 1 = -3$.

  • For $x = 3$: since $3 \ge 0$, evaluate $f(3) = (3)^2 - 3 = 6$.

Domain and Range

• Domain: The set of all valid input values ($x$-values) for which the function is defined.

  • Algebraic Restrictions:

  • Division by zero is undefined: If $f(x) = \frac{1}{g(x)}$, set $g(x) \neq 0$.

  • Radicands of even roots must be non-negative: If $f(x) = \sqrt{g(x)}$, set $g(x) \ge 0$.

• Range: The set of all resulting output values ($y$-values or $f(x)$) produced over the entire domain.

• Interval Notation Rules:

  • Use brackets $[a, b]$ when values are included ($\le$ or $\ge$).

  • Use parentheses $(a, b)$ when values are excluded ($<$ or $>$), and always for negative infinity $(-\infty)$ and positive infinity $(\infty)$.

  • Use the union symbol $\cup$ to join disjoint intervals (e.g., $(-\infty, 2) \cup (2, \infty)$).

Key Features of Functions

• Zeros ($x$-Intercepts): The values of $x$ for which $f(x) = 0$. Points are written as $(x, 0)$.

• $y$-Intercept: The output value where the function crosses the vertical axis, found by evaluating $f(0)$. Points are written as $(0, f(0))$.

• Extrema (Maximum / Minimum):

  • Relative (Local) Extrema: Peaks or valleys occurring within an open interval around a point.

  • Absolute (Global) Extrema: The highest or lowest value over the entire domain of the function.

• Intervals of Increase, Decrease, and Constant:

  • Intervals are always described using open intervals of $x$-values:

  • Increasing: The graph rises from left to right ($f(x_1) < f(x_2)$ when $x_1 < x_2$).

  • Decreasing: The graph falls from left to right ($f(x_1) > f(x_2)$ when $x_1 < x_2$).

  • Constant: The graph is horizontal ($f(x_1) = f(x_2)$).

• Sign of a Function (Positive / Negative):

  • Positive: $f(x) > 0$ (intervals where the graph lies strictly above the $x$-axis).

  • Negative: $f(x) < 0$ (intervals where the graph lies strictly below the $x$-axis).

• Even and Odd Functions (Symmetry):

  • Even Function: Satisfies $f(-x) = f(x)$. Graph is symmetric with respect to the $y$-axis.

  • Odd Function: Satisfies $f(-x) = -f(x)$. Graph is symmetric with respect to the origin ($180^\circ$ rotational symmetry).

• Linear Functions & Real-World Graphs:

  • Standard form: $f(x) = mx + b$, where $m$ represents the constant rate of change (slope) and $b$ represents the initial value ($y$-intercept).

  • In applied contexts, identify continuous vs. discrete data, interpret slopes as rates (e.g., change in output per unit input), and identify physical boundaries on domain and range.

2.2 Composition and Combination of Functions

Transformations of Functions

Given the standard transformation form $g(x) = a \cdot f(b(x - h)) + k$:

• Vertical Shifts:

  • $g(x) = f(x) + k$ shifts upward by $k$ units ($k > 0$).

  • $g(x) = f(x) - k$ shifts downward by $k$ units ($k > 0$).

• Horizontal Shifts:

  • $g(x) = f(x - h)$ shifts rightward by $h$ units ($h > 0$).

  • $g(x) = f(x + h)$ shifts leftward by $h$ units ($h > 0$).

• Reflections:

  • Vertical reflection across the $x$-axis: $g(x) = -f(x)$

  • Horizontal reflection across the $y$-axis: $g(x) = f(-x)$

• Vertical Dilations:

  • Vertical stretch by a factor of $|a|$ when $|a| > 1$.

  • Vertical compression (shrink) by a factor of $|a|$ when $0 < |a| < 1$.

• Horizontal Dilations:

  • Horizontal compression by a factor of $\frac{1}{|b|}$ when $|b| > 1$.

  • Horizontal stretch by a factor of $\frac{1}{|b|}$ when $0 < |b| < 1$.

Combinations of Functions

For two functions $f(x)$ and $g(x)$ with domains $D_f$ and $D_g$:

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(f \cdot g)(x) = f(x) \cdot g(x)$

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

• Domain of Combined Functions: The domain consists of all $x$ belonging to $D_f \cap D_g$, with the additional restriction that $g(x) \neq 0$ for quotients.

Composition of Functions

• Definition: Applying one function to the result of another function:   $(f \circ g)(x) = f(g(x))$

• Evaluation: Substitute the expression or value of $g(x)$ into every variable instance of $f(x)$.

• Domain of Composite Function: The set of all inputs $x$ in the domain of $g$ such that $g(x)$ is also contained in the domain of $f$.

Inverse Functions

• Definition: A function $f^{-1}(x)$ is the inverse of $f(x)$ if it reverses the mapping:   $f(a) = b \iff f^{-1}(b) = a$

• One-to-One and the Horizontal Line Test: A function has an inverse function if and only if it passes the horizontal line test (no horizontal line intersects the graph at more than one point).

• Finding an Inverse Algebraically:

  1. Replace the function notation $f(x)$ with $y$.

  2. Interchange the variables $x$ and $y$.

  3. Solve the resulting equation explicitly for $y$.

  4. Rename the variable $y$ as $f^{-1}(x)$.

• Verifying Inverses: Two functions $f(x)$ and $g(x)$ are proved to be inverses if and only if both conditions hold:   $(f \circ g)(x) = x \quad \text{and} \quad (g \circ f)(x) = x$

• Graphical Properties:

  • The graph of $f^{-1}(x)$ is a reflection of $f(x)$ across the line $y = x$.

  • The domain of $f(x)$ equals the range of $f^{-1}(x)$, and the range of $f(x)$ equals the domain of $f^{-1}(x)$.