Library of Functions You should be familiar with the shapes of these basic functions. We'll learn them in this section.

Linear Functions Equations that can be written f(x) = mx + b slope y-intercept The domain of these functions is all real numbers.

Constant Functions f(x) = b, where b is a real number Would constant functions be even or odd or neither? The domain of these functions is all real numbers. The range will only be b

Identity Function f(x) = x f(x) = x, slope 1, y-intercept = 0 If you put any real number in this function, you get the same real number “back”. f(x) = x Identity Function f(x) = x, slope 1, y-intercept = 0 Would the identity function be even or odd or neither? Continuous Increasing (-∞, ∞) The domain of this function is all real numbers. The range is also all real numbers

Square Function f(x) = x2 Would the square function be even or odd or neither? Continuous Bounded below Increasing [0, ∞] Decreasing (-∞ , 0] The domain of this function is all real numbers. The range is all NON-NEGATIVE real numbers

Cube Function f(x) = x3 Would the cube function be even or odd or neither? Continuous Increasing for all x Not bounded above or below No local extrema No Horizontal asymptotes No vertical asymptotes End behavior: As x approaches -∞ the function = -∞. As x approaches ∞ the function = ∞ The domain of this function is all real numbers. The range is all real numbers

Square Root Function Would the square root function be even or odd or neither? Continuous on [0, ∞] Increasing on [0, ∞] Bounded below but not above Local minimum at x = 0 No horizontal asymptotes No vertical asymptotes End behavior: As x approaches ∞ the square root function = ∞ The domain of this function is NON-NEGATIVE real numbers. The range is NON-NEGATIVE real numbers

Reciprocal Function The domain of this function is all NON-ZERO real numbers. Would the reciprocal function be even or odd or neither? Not continuous, has an infinite discontinuity at x = 0 The range is all NON-ZERO real numbers.

Absolute Value Function The domain of this function is all real numbers. Would the absolute value function be even or odd or neither? Continuous Decreasing on (∞, 0]; increasing on [0, ∞) Bounded below Local minimum at (0, 0) No horizontal asymptotes No vertical asymptotes End behavior: As x approaches -∞ |x| = ∞ and as x approaches ∞ |x| = ∞ The range is all NON-NEGATIVE real numbers

Exponential Function The range of this function is {y| y > 0 } f(x) = Would the exponential function be even or odd or neither? Exponential Function The domain of this function is (-∞, ∞) Continuous Increasing for all x Bounded below, but not above No local extrema Horizontal asymptote: y = 0 No vertical asymptotes End behavior: As x approaches -∞ e^x = 0 As x approaches ∞ e^x = ∞ The range of this function is {y| y > 0 }

f(x) = ln x Logarithmic Graph Would the absolute value function be even or odd or neither? Logarithmic Graph The domain of this function is {x| x > 0} Continuous on (0, ∞) Increasing on (0, ∞) No symetry Not bounded above or below No local extrema No horizontal asymptotes Vertical asymptote: x = 0 End behavior: As x approaches ∞ ln x = ∞ The range of this function is (-∞,∞)

Sine Function The domain of this function is all real numbers f(x) = sin x Would the sine function be even or odd or neither? Sine Function The domain of this function is all real numbers Continuous Alternately increasing and decreasing in periodic waves Symmetric with respect to the origin (odd) Bounded Absolute maximum of 1 Absolute minimum of -1 No horizontal asymptotes No vertical asymptotes End behavior: as the limit of x approaches -∞ of sin x and the limit of x approaches ∞ sinx the limit does not exist The range of this function is [ -1, 1]

Cosine Function f(x) = cos x Would the cosine function be even or odd or neither? Cosine Function Continuous Alternately increasing and decreasing in periodic waves Bounded Absolute maximum 1 Absolute minimum of -1 No Horizontal asymptotes No Vertical asymptotes End behavior: As the limit approaches -∞ and the limit approaches ∞ cos x do not exist. (The function values continually oscillate between -1 and 1 and approach no limit.) The domain of the function is all real numbers The range of this function is [-1, 1]

The Greatest integer Function f(x) = int (x) Would the greatest integer function be even or odd or neither? The Greatest integer Function Discontinuity at every integer value of x. It is not a continuous function Increasing/Decreasing behavior: constant on intervals of the form [k, k+1], where k is an integer; No boundedness Local extrema: every non-integer is both a local minimum and local maximum; Horizontal asymptotes: none Vertical asymptotes: none; End behavior: int (x) = -∞ As x approaches -∞ and int(x) = ∞ As x approaches ∞. The domain of this function is all real numbers The range of this function is all integers

The Logistic Function The domain of this function is all real numbers f(x) = Would the logistic function be even or odd or neither? The Logistic Function The domain of this function is all real numbers Continuous Increasing for all x Bounded below and above Symmetric about (0, ½), but neither even nor odd No local extrema Horizontal asymptotes: y = 0 and y = 1 No vertical asymptotes End behavior: as x approaches -∞ f(x) = 0 and as x approaches ∞ f(x) = 1 The range of this function is 0 < y <1

These are functions that are defined differently on different parts of the domain. WISE FUNCTIONS

This then is the graph for the piecewise function given above. This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x2 What does the graph of f(x) = -x look like? What does the graph of f(x) = x2 look like? Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0. Remember y = f(x) so lets graph y = x2 which is a square function (parabola) Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0. Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph. This then is the graph for the piecewise function given above.

For x > 0 the function is supposed to be along the line y = - 5x. For x = 0 the function value is supposed to be –3 so plot the point (0, -3) For x values between –3 and 0 graph the line y = 2x + 5. Since you know the graph is a piece of a line, you can just plug in each end value to get the endpoints. f(-3) = -1 and f(0) = 5 Since you know this graph is a piece of a line, you can just plug in 0 to see where to start the line and then count a – 5 slope. open dot since not "or equal to" So this the graph of the piecewise function solid dot for "or equal to"