PRE CAL INTRO

Functions: Graphs, Domain, and Range

Basic Parent Functions

• Functions Overview: Understanding functions, their graphs, domains, and ranges is crucial in pre-calculus.

• Domain: Represents the set of all possible x-values that a function can accept.

  • Determined by analyzing the function from left to right on its graph.

• Range: Represents the set of all possible y-values that a function can output.

  • Determined by analyzing the function from bottom to top on its graph.

• Interval Notation: When writing domain and range, use parentheses for values that are not included (e.g., infinity) and brackets for values that are included.

Linear Function: y = x

• Graph: A straight line passing through the origin (0, 0).

• Domain: All real numbers, (-\infty, \infty). X can be any value.

• Range: All real numbers, (-\infty, \infty). Y can be any value.

Quadratic Function: y = x^2

• Graph: A parabola, shaped like an upward U.

• Domain: All real numbers, (-\infty, \infty).

• Range: From 0 to infinity, [0, \infty). The lowest y-value is 0, and it includes 0.

Cubic Function: y = x^3

• Graph: An increasing function that extends in both positive and negative directions.

• Domain: All real numbers, (-\infty, \infty).

• Range: All real numbers, (-\infty, \infty).

Square Root Function: y = \sqrt{x}

• Graph: Increases at a decreasing rate, starting from the origin and extending to the right.

• Domain: From 0 to infinity, [0, \infty). It includes 0 because the square root of 0 is 0.

• Range: From 0 to infinity, [0, \infty).

Cube Root Function: y = \sqrt[3]{x}

• Graph: Similar to the square root function but symmetric about the origin, extending in both positive and negative directions.

• Domain: All real numbers, (-\infty, \infty).

• Range: All real numbers, (-\infty, \infty).

• Note: Unlike the square root function, you can plug in negative values for x in the cube root function, and the result will be a real number (e.g., the cube root of -8 is -2).

Absolute Value Function: y = |x|

• Graph: Forms a V-shape, opening upwards.

• Domain: All real numbers, (-\infty, \infty).

• Range: From 0 to infinity, [0, \infty). The lowest y-value is 0.

Rational Function: y = \frac{1}{x}

• Asymptotes: Has a horizontal asymptote at y = 0 and a vertical asymptote at x = 0.

• Graph: Symmetric about the origin.

• Domain: All real numbers except 0, (-\infty, 0) \cup (0, \infty). The vertical asymptote must be excluded.

• Range: All real numbers except 0, (-\infty, 0) \cup (0, \infty). The horizontal asymptote must be excluded.

Rational Function: y = \frac{1}{x^2}

• Asymptotes: Has a horizontal asymptote at y = 0 and a vertical asymptote at x = 0.

• Graph: Symmetric about the y-axis (reflection across the y-axis). The function can only have positive values.

• Domain: All real numbers except 0, (-\infty, 0) \cup (0, \infty).

• Range: From 0 to infinity, but not including 0, (0, \infty). The function gets very close to zero but never touches it (y=0).

  • As x increases (e.g., 1, 100, 1000), y gets closer to zero (e.g., 1, 0.01, 0.000001) but never reaches it.

Exponential Function: y = e^x

• Asymptote: Has a horizontal asymptote at y = 0.

• Graph: Increases at an increasing rate.

• Domain: All real numbers, (-\infty, \infty).

• Range: From 0 to infinity, but not including 0, (0, \infty).

Natural Log Function: y = \ln(x)

• Inverse Function: The inverse of the exponential function. The domain and range are switched compared to the exponential function.

• Asymptote: Has a vertical asymptote at x = 0.

• Graph: Increases at a decreasing rate.

• Domain: From 0 to infinity, but not including 0, (0, \infty).

• Range: All real numbers, (-\infty, \infty).

• Symmetry: The graphs of e^x and \ln(x) are symmetrical across the line y = x, which is typical of inverse functions.

Trigonometric Functions: Sine, Cosine, and Tangent

Sine Function: y = \sin(x)

• Graph: A periodic wave-like function that starts from the origin.

• Amplitude: The number in front of the sine function (e.g., 1 for a standard sine function), which determines the height of the wave from its center line.

• Domain: All real numbers, (-\infty, \infty).

• Range: From -1 to 1, [-1, 1] if the amplitude is 1.

Cosine Function: y = \cos(x)

• Graph: A periodic wave-like function that starts at the top (maximum y-value) instead of the origin.

• Amplitude: Similar to the sine function.

• Domain: All real numbers, (-\infty, \infty).

• Range: From -1 to 1, [-1, 1] if the amplitude is 1.

Tangent Function: y = \tan(x)

• Asymptotes: Has vertical asymptotes at x = \frac{n\pi}{2}, where n is an odd integer (e.g., -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}).

• Graph: An increasing function that repeats across each vertical asymptote.

• Domain: All real numbers except the vertical asymptotes, so X cannot be \frac{n\pi}{2} where n is an odd integer.

• Range: All real numbers, (-\infty, \infty).

Transformations

• Parent Function: A basic function (e.g., f(x)) that can be transformed to create other functions.

• Vertical Stretch: y = k \cdot f(x), where k > 1. Stretches the graph vertically by a factor of k, doubling the y-values.

• Vertical Shrink: y = \frac{1}{2} \cdot f(x). Shrinks the graph vertically by a factor of 2, halving the y-values.

• Horizontal Shrink: y = f(2x). Shrinks the graph horizontally by a factor of 2, halving the x-values.

• Horizontal Stretch: y = f(\frac{1}{2} x). Stretches the graph horizontally by a factor of 2, doubling the x-values.

• Shift Right: y = f(x - 4). Shifts the graph 4 units to the right.

  • Set the inside term equal to zero and solve for x.

• Shift Left: y = f(x + 3). Shifts the graph 3 units to the left.

• Reflect over x-axis: y = -f(x). Flips the graph over the x-axis.

• Reflect over y-axis: y = f(-x). Flips the graph over the y-axis.

• Combined Reflection. y = -f(-x). Reflects over the origin which is the combination of reflecting over both the x and y axis

Inverse Functions

• Switching Coordinates. To graph and inverse function switch the x and y coordinates.

• Reflection across y=x. Inverse functions are reflected across the line y=x

Graphing Techniques and Transformations: Additional Examples

Vertical Shift: y = x^2 - 3

• Transformation: The function is shifted downwards by 3 units.

• Domain: No change as its all real numbers, (-\infty, \infty). This is true for all qudratic functions.

• Range: Changed, since shifted downwards. Now it's [-3, \infty). Minimum y values is -3.

Vertical Shift and Reflection. y = -(x^2 + 2)

• Transformation: The function is shifted upwards by 2 units as well as reflected about the x axis(since the leading term is negative). Because the expression is -x^2, the parabola opens downwards.

• Domain: No change as its all real numbers, (-\infty, \infty).

• Range: Changed, since shifted upwards but then reflected about the x axis. The rage is now (-\infty, 2]. Maximum y value is 2.

Horizontal Shift: y = (x - 2)^3

• Transformation: The function is shifted right by 2 units.

  • The new origin is located at positive 2.

• Domain: No change as its all real numbers, (-\infty, \infty). This is what the domain will be for all cubic functions as well.

• Range: No change as its all real numbers, (-\infty, \infty).

Rational Functions and Horizontal Shift: y = \frac{1}{x-3}

• Transformation: The function is shifted right by 3 units.

  • Since X cannot be zero, because the denominator in the expression cannot equal 0, then x \neq 3 since x-3 cannot be 0

  • Set the denominator equal to zero to determine the new vertical asymptote.

• Domain: Changed! It is what x can be, which in this case is everyting but 3. (-\infty, 3) \cup (3, \infty).

• Range: No change because of the horizontal shift. The function can still be any value besides 0. (-\infty, 0) \cup (0, \infty).

Rational Functions, Horizontal Shift and Vertical Shift. y = \frac{1}{x} + 2

• Transformation: The function is shifted upwards by 2 units.

  • x \neq 0 since the denominator in the expression cannot equal 0. A vertical asymptote exists at that exclusion

  • Horizontal Asymptote. The horizontal asymptote shifted upwards two units, to +2

• Domain: No change because of the vertical shift. (excluding the asymptote). (-\infty, 0) \cup (0, \infty).

• Range: Changed! It is what y can be, which in this case is everyting but 2. (-\infty, 2) \cup (2, \infty).

Rational Functions, Horizontal and Vertical Shift and Reflections about the x axis. y = -\frac{1}{x+2} + 3

• Transformation: Horizontal Shift left 2 units, vertical shift upwards by 3 units, and reflection about the x axis(due to negative sign).

  • x \neq -2 since the denominator in the expression cannot equal 0. A vertical asymptote exists at that exclusion

    • The negative sign causes the graph to reflect about the horizontal asymptote, with the general shape oriented differently than before.

• Domain: Changed! It is what x can be, which in this case is everything but -2. (-\infty, -2) \cup (-2, \infty).

• Range: Changed! All values, exception for 3. (-\infty, 3) \cup (3, \infty).

Rational Functions, Horizontal and Vertical Shift, Reflections about the x axis and exponent component in the denominator. y = \frac{1}{(x-2)^2} + 3

• Transformation:The function is shifted right 2 units, shift upwards by 3 units. The exponent leads to it being symetrical about the new horizontal asymptote (since x cannot be 2).

  • Vertical Asymptote at x=2 and Horizontal Asymptote at y=3

• Domain: Changed! It is what x can be, which in this case is everything but 2. (-\infty, -2) \cup (-2, \infty).

• Range: Changed! Since it is the exponent is a perfect square, it will only be values y \ge 3. Since, as shown before, the asymptote is excluded. (3, \infty).

Rational Functions, Horizontal and Vertical Shift, Reflections about the x axis and an overall negative sign(outside of the expression) y = -\frac{1}{(x+3)^2} - 2

• Transformation:Horizontal Shift left 3 units Vertical shift Downwards 2 units and Reflections about the Horizontal vertical asyptote, and horizontal asymptote.

  • Vertical Asymptote = -3 and Horizontal Asymptote = -2

• Domain: (-\infty, -3) \cup (-3, \infty). X can be everything but -3

• Range: Since the perfect square is in the denominator and there is a negative sign overall. The values are all y \le -2 so it's written as (-\infty, -2)

Absolute Value Functions, Horizontal/Vertial shift, y= |x-3| + 1

• The graph will have a standard V slope, the main point will simply move upwards and to the right or left depending on the numbers.

  • With the shift described. The new point will be (3,1) since those are the zero values.

  • Domain: (-\infty, +\infty)

  • Range:[1, +\infty)

Absolute Value Functions, Horizontal/Vertial shift+negative sign, y= -|x-2| + 2

• The graph will have a inverted V slope (due to negative slop)e, the main point will simply move upwards and to the right or left depending on the numbers.

  • With the shift described. The new point will be (2,2) since those are the zero values. Not that due to the orientation of the graph. Two will be the maximum point instead of the minimum.

  • Domain: (-\infty, +\infty)

  • Range: (-\infty, 2]

Exponential Functions y = e^{x+2}

• With the horizontal asymptote y=0. the value of y will always be greater than it unless stated otherwise, in which case the asymptote can potentially shift

  • Horizontal Asymptote will shift uwpards by 2. to y=2

  • Domain: (-\infty, +\infty)

  • Range: (2, +\infty)

Logarithmic Functions y \ln(x + 3)

• The vertical asymptote will shift depending on the formula, in this case, since its \ln(x + 3). It will be -3, since that is the value that can't be exeed as the log function doesnt support negative numbers

  • Domain: (-3, +\infty)

  • Range: (-\infty, +\infty)

Transformed Trignometric Equations y = 2\cdot sin(x) + 1

• Trig functions usually oscilate in a defined manor based on their range, for sin its usually (-1,1). However you can transofrm it using constant values or by adding multiplying terms.

  • This function will have sine wave with an amplitude of 2, shifted upwards by 1 unit. Maximum being 3 and miminum value will be -1

• Domain: (-\infty, +\infty)

• Range: [-1,3]$$

Compositie Funtions: General

Defining composite functions

• You can have one compositie funtion, inserterd inside another

• Denoted as something like f(g(x)) or other letters

• With compositie functions that can be simplified, you can have g(f(x)), meaning you input the f(x) function into the original g(x) function

• In the case with a defined intput value such as g(f(3)), then you input 3 into the function, and that output is entered into the g() function. To achieve a defined number outputted.

• The same concept applies in reverse with f(g(2)). Whatever the output is for G is put into the F function.

Inverse functions: General Concepts

• Inverse functions invert the inputs and outputs and can be symetric. If they are symmetric there should be a line where the functions can be reflected about.

• If the function is denoted y in that case; you replace the y with x, and then replace the x after with a y. From there you simiplify. This is the process to create an inverse formula