我在学习微积分，这是教材内容，请你仔细讲解一下相关内容。---**Textual Information:** * **Title:** 함수 (Function) * **Heading:** 함수의 그래프 (Graph of a function) * **Definition:** 함수 $y = f(x)$ 의 그래프는 순서쌍의 집합: $\{(x, f(x)) | x \in X \}$ (The graph of the function $y = f(x)$ is the set of ordered pairs: $\{(x, f(x)) | x \in X \}$) **Chart/Diagram Description:** **Chart 1:** * **Type:** Cartesian coordinate system showing a function's graph. * **Coordinate Axes:** X-axis and Y-axis intersecting at the origin (labeled '0'). Arrows indicate the positive direction of both axes. * **Curve:** A smooth, blue curve representing the graph of a function. * **Points:** * A point is labeled with coordinates `(x, f(x))` on the curve. * **Lines/Segments:** * Vertical dashed lines connect points on the x-axis (1, 2, x) to the curve. * Vertical brackets are used to indicate the height of the function at specific x-values. * **Labels and Annotations:** * The curve is implicitly represented as `y = f(x)`. * Labels `f(1)` and `f(2)` with brackets indicate the function's value (y-coordinate) at x=1 and x=2, respectively. These brackets are pink and purple. * A label `f(x)` with a bracket indicates the function's value (y-coordinate) at x, represented by a vertical segment from the x-axis at x up to the point `(x, f(x))` on the curve. **Chart 2:** * **Type:** Cartesian coordinate system showing the domain, codomain, and range of a function. * **Coordinate Axes:** X-axis and Y-axis intersecting at the origin (labeled '0'). Arrows indicate the positive direction of both axes. * **Curve:** A smooth, blue curve labeled `y = f(x)`. The curve is bounded by dashed lines forming a rectangle. * **Lines/Segments:** * Horizontal dashed lines extend from the endpoints of the curve to the Y-axis. * Vertical dashed lines extend from the endpoints of the curve to the X-axis. * A bracket on the X-axis is labeled `정의역` (Domain), representing the interval of x-values for which the function is defined. This bracket is red. * A bracket on the Y-axis is labeled `치역` (Range), representing the set of y-values that the function actually takes for the given domain. This bracket is green. * A larger bracket on the Y-axis is labeled `공역` (Codomain), representing the set of possible y-values that the function could output. This bracket is blue. * **Labels and Annotations:** * `y = f(x)` labels the curve. * `정의역` (Domain) is labeled below the X-axis, indicating the set of input values. * `치역` (Range) is labeled to the left of the Y-axis, indicating the set of output values achieved by the function. * `공역` (Codomain) is labeled to the right of the Y-axis, indicating a set containing the range (possible output values). * Points are marked on the X and Y axes corresponding to the boundaries of the domain and range/codomain. The boundary points on the axes are colored red, green, and blue corresponding to the respective labels. **Title:** 함수 (Function) **Section Title:** 예제 (Example) **Question Stem:** - 함수 $f$ 의 그래프가 그림 6과 같다고 하자. (Let's assume the graph of function $f$ is as shown in Figure 6.) **(a)** $f(1)$ 의 값을 구하여라. (Find the value of $f(1)$.) **(b)** $f$ 의 정의역과 치역을 구하여라. (Find the domain and range of $f$.) **Chart Description:** * **Type:** Line chart representing the graph of a function. * **Main Elements:** * **Coordinate Axes:** A Cartesian coordinate system with a horizontal X-axis labeled 'x' and a vertical Y-axis labeled 'y'. The origin (0,0) is marked. The positive X-axis points right, and the positive Y-axis points up. * **Scale:** The grid lines indicate a scale. On both the X and Y axes, the major grid lines are spaced by 1 unit, starting from 0. The number '1' is explicitly labeled on both the positive X and positive Y axes. * **Graph:** A smooth curve (presumably the graph of the function $f$) is drawn in magenta color. The curve starts from a point on the Y-axis at (0, 1). It rises to a peak, then decreases, crosses the X-axis, reaches a minimum, and then rises again to end at a point on the X-axis. * **Specific Points:** * The graph starts at (0, 1). * The graph passes through the point where x=1. Observing the graph, at x=1, the y-value appears to be approximately 2. * The graph appears to cross the x-axis at approximately x = 2.5. * The graph appears to have a local minimum at approximately x = 3.5, with a y-value slightly below 0. * The graph ends on the x-axis at x=4. (The rightmost point of the graph is at x=4, y=0). * **Labels:** The chart is labeled "그림 6" (Figure 6) below the X-axis. * **Domain and Range (visual estimation from graph):** * The graph extends horizontally from x=0 to x=4. * The graph extends vertically from a minimum value (slightly below 0, near the point (3.5, y_min)) to a maximum value (near the peak, approximately at x=1.5, y_max, where y_max is approximately 2.5). The graph also includes the points (0,1) and (4,0). Looking closely, the minimum y-value is between 0 and -1, and the maximum y-value is between 2 and 3. The lowest point seems to be slightly below the x-axis, and the highest point seems to be above y=2. **Other Relevant Text:** 그림 6 (Figure 6) **Title:** 함수 **Section Title:** Example **Problem 1:** - 함수 y = x^2의 정의역과 치역을 말하고 그래프를 그려라. (Translation: State the domain and range of the function y = x^2 and draw the graph.) **Problem 2:** - 함수 y = $\sqrt{1 - x^2}$의 정의역과 치역을 말하고 그래프를 그려라. (Translation: State the domain and range of the function y = $\sqrt{1 - x^2}$ and draw the graph.) **Text Extraction:** 조각함수 * 조각함수란? - 정의역의 각 부분에서 서로 다른 수식으로 정의되는 함수를 조각함수(piecewise function) 라 한다. Title: 조각함수 Section: 예제 Text: 함수 f는 Mathematical Formula: $f(x) = \begin{cases} 1 - x, & x \le -1 \\ x^2, & x > -1 \end{cases}$ Text: 로 정의 되어 있다. Question: $f(-2), f(-1)$ 및 $f(0)$의 값을 구하시오. 조각함수 • 예제 - 절댓값 함수 f(x) = |x|의 그래프를 그려라. **Title:** 우함수와 기함수 **Content:** * **우함수란?** - 함수 f가 정의역에 있는 모든 x에 대하여 $f(-x) = f(x)$를 만족하면, f를 우함수(짝수함수, even function)라 한다. - 예를 들어, - 함수 $f(x) = x^2$은 우함수 $f(-x) = (-x)^2 = x^2 = f(x)$ **Chart/Diagram Description:** * **Type:** Cartesian coordinate plane graph showing a function. * **Coordinate Axes:** * X-axis labeled 'x'. * Y-axis labeled 'y'. * Origin labeled '0'. * **Graph:** A blue curve representing a function, which appears to be a parabola opening upwards, symmetric about the y-axis. * **Points and Labels:** * Two points are marked on the curve at x-coordinates -x and x. * Horizontal lines extend from these points to the y-axis level, indicating the function values at -x and x. * Labels "f(-x)" and "f(x)" are placed vertically next to the lines extending from the points at -x and x respectively, indicating the y-values. * Braces `{}` are used to indicate the height corresponding to f(-x) and f(x). * Equal signs `=` are shown within the braces, suggesting that $f(-x) = f(x)$. * Pink horizontal lines and tick marks are drawn between the points at x and -x on the y=f(x) level, and on the x-axis, illustrating the x and -x positions. * **Annotations:** * 그림 19 (Figure 19) ```plaintext 우함수와 기함수 * **기함수란?** - 함수 f가 정의역에 있는 모든 x에 대하여 f(-x) = -f(x)를 만족하면, f를 기함수(홀수함수, odd function)라 한다. - 예를 들어, - 함수 f(x) = x³은 기함수 f(-x) = (-x)³ = -x³ = -f(x) - 기함수의 그래프는 원점에 대하여 대칭이다. **Chart/Diagram Description:** * **Type:** Coordinate plane graph showing the graph of an odd function. * **Coordinate Axes:** X-axis labeled 'x' and Y-axis labeled 'y'. The origin (0) is marked. * **Curve:** A blue curve passing through the origin, extending into the first and third quadrants, showing rotational symmetry about the origin. * **Points:** - A point on the curve in the first quadrant with x-coordinate 'x' and y-coordinate 'f(x)'. - A point on the curve in the third quadrant with x-coordinate '-x'. * **Lines:** - Vertical dashed lines from the points on the curve to the x-axis. - Pink line segments connecting the origin (0,0) to the points on the curve. These segments are marked with single hash marks, indicating equal length. - Vertical segments from the x-axis up to the points on the curve are marked with double hash marks on the left side point's vertical segment and its corresponding y-value on the left y-axis implied by the point's position. The right side vertical segment from x to f(x) is labeled with '{ f(x)'. The double hash marks on the vertical segments suggest |f(-x)| = |f(x)|. * **Labels and Annotations:** - X-axis: -x, 0, x - Y-axis: y - Point in Q1: Labeled with associated f(x) height using a curly brace. - Figure caption below the graph: 그림 20 (Figure 20) * **Relative Position and Direction:** The graph visually demonstrates that for a point (x, f(x)) on the curve, the point (-x, -f(x)) is also on the curve, which is equivalent to (-x, f(-x)) = (-x, -f(x)) or f(-x) = -f(x). The point (-x, f(-x)) is shown in the third quadrant, and the visual symmetry about the origin is highlighted by the pink lines from the origin to the points and the similar triangles formed. The double hash marks on the vertical lines further suggest the relationship between f(x) and f(-x). ``` Title: 우함수와 기함수 (Even Functions and Odd Functions) Section Heading: 예제 (Example) Instruction: - 다음 함수가 우함수인지, 기함수인지, 또는 둘 다 아닌지를 결정하여라. (Determine whether the following functions are even, odd, or neither.) Functions: (a) $f(x) = x^5 + x$ (b) $g(x) = 1 - x^4$ (c) $h(x) = 2x - x^2$