Show pageOld revisionsBacklinksFold/unfold allBack to top This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== Pre-AP Calculus ====== Book: PRECALCULUS Mathematics for Calculus by James Stewart, Lothar Redlin, Saleem Watson. ===== Chap.2 Function ===== ==== What is a Function? ==== > Function Definition > A function \( f \) assigns to each element \( x \) in a set \( A \) exactly one element, denoted as \( f(x) \), in a set \( B \). Following is not a function. {{url>https://www.desmos.com/calculator/pkb2vug0cf?embed 500,500 fullscreen | This is not a function}} === Evaluating a Function === \[ \frac{f(x + h) - f(x)}{h}, \quad h \neq 0 \] ### Step 1: Compute \( f(x + h) \) \[ f(x + h) = 2(x + h)^2 + 3(x + h) - 1 \] \[ = 2(x^2 + 2xh + h^2) + 3x + 3h - 1 \] \[ = 2x^2 + 4xh + 2h^2 + 3x + 3h - 1 \] ### Step 2: Compute \( f(x + h) - f(x) \) \[ f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 + 3x + 3h - 1\right) - \left(2x^2 + 3x - 1\right) \] \[ = 4xh + 2h^2 + 3h \] ### Step 3: Form the Difference Quotient \[ \frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 + 3h}{h} \] \[ = \frac{h(4x + 2h + 3)}{h} \] \[ = 4x + 2h + 3 \quad \text{(for \( h \neq 0 \))} \] === The Domain of a Function === > Recall that the //domain// of a function is the set of all inputs for the function. If the function is given by an algebraic expression and the domain is not stated explicitly, then by //convention the domain of the function is the domain of the algebraic expression—that is, the set of all real numbers for which the expression is defined as a real number.// === Four Ways to Represent a Function === - verbally (by a description in words) - algebraically (by an explicit formula) - visually (by a graph) - numerically (by a table of values) ==== Graphs of Functions ==== The most important way to visualize a function is through its graph. In this section we investigate in more detail the concept of graphing functions. === Graphing Functions === === Graphing Piecewise Defined Functions === \[ f(x) = \begin{cases} x^2 & \text{if } x \leq 1, \\ 2x + 1 & \text{if } x > 1. \end{cases} \] ==== Increasing and Decreasing Functions; Average Rate of Change ==== Functions are often used to model changing quantities. In this section we learn how to determine if a function is increasing or decreasing, and how to find the rate at which its values change as the variable changes. === Increasing and Decreasing Functions === **Increasing Function:** A function \( f \) is increasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \leq f(x_2). \] **Strictly Increasing Function:** A function \( f \) is strictly increasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) < f(x_2). \] **Decreasing Function:** A function \( f \) is decreasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \geq f(x_2). \] **Strictly Decreasing Function:** A function \( f \) is strictly decreasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) > f(x_2). \] === Average Rate of Change === Definition: For a function \( f(x) \), the average rate of change over the interval \([a, b]\) is given by: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] This formula calculates the change in the function's value (\( \Delta y \)) divided by the change in the input (\( \Delta x \)). ==== Transformations of Functions ==== In this section we study how certain transformations of a function affect its graph. This will give us a better understanding of how to graph functions. The transformations we study are shifting, reflecting, and stretching. === Vertical Shifting === Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if it is negative. Use the graph \( f(x) = x^2 \) of to sketch the graph of each function. \[ a.\quad g(x) = x^2 + 3 \\ b.\quad h(x) = x^2 - 2 \] {{url>https://www.desmos.com/calculator/uoacsjlmc5?embed 500,500}} === Horizontal Shifting === **Shift Right by \( h \) Units** \[ y = f(x - h) \] **Example:** \( f(x) = x^2 \) shifted right by 2: \[ y = (x - 2)^2 \] **Shift Left by \( h \) Units** \[ y = f(x + h) \] **Example:** \( f(x) = \sqrt{x} \) shifted left by 3: \[ y = \sqrt{x + 3} \] **Key Rule** **\( -h \) inside \( f(x - h) \) → shifts right** **\( +h \) inside \( f(x + h) \) → shifts left** === Reflecting Graphs === To graph \( y = -f(x) \), reflect the graph of \( y = f(x) \) in the x-axis. To graph \( y = f(-x) \), reflect the graph of \( y = f(x) \) in the y-axis. === Vertical Stretching and Shrinking === To graph \( y = cf(x) \), If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of c. If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of c. === Horizontal Stretching and Shrinking === To graph \( y = f(cx) \), If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of 1/c. If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of 1/c. === Even and Odd Functions === If a function \( f \) satisfies \( f(x) = f(-x) \) for every number \( x \) in its domain, then \( f \) is called an even function. If \( f \) satisfies \( f(x) = -f(-x) \) for every number \( x \) in its domain, then \( f \) is called an odd function. ==== Quadratic Functions; Maxima and Minima ==== math/pre-ap_calculus.txt Last modified: 2025/05/14 16:47by root1
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