Functions from a Calculus Perspective Chapter One

Functions, Domain and Range Function: A function is a relation between two variables such that the output can be predicted for every input in the domain. Domain: The set of possible inputs (usually x-values) Range: The set of possible outputs (usually y-values) Different inputs can have the same outputs (y-values can repeat) One input cannot have multiple outputs (x-values cannot repeat) The Vertical Line Test is used to determine if a graph represents a function.

Identifying Domain and Range from a Graph Domain: x-values used in drawing the graph [leftmost value, rightmost value] minus any points of discontinuity Range: y-values produced when drawing a graph [lowest value, highest value] minus any points of discontinuity

Intercepts y-intercept: x-intercept(s), aka zero(s) of the function,: point where the graph crosses the y-axis set x = 0: (0, ) x-intercept(s), aka zero(s) of the function,: point where the graph crosses the x-axis set f(x)=0: ( , 0)

Even and Odd Functions Even function: Odd function: symmetric to y-axis y-axis acts like a mirror (left side is a mirror image of the right side) f(-x) = f(x) replace x with –x to get f(-x) then simplify compare to f(x), if same then even If (a, b) is on the graph of f(x) then (-a, b) will also be on the graph. Odd function: symmetric to origin origin acts like the axle of a wheel (graph looks the same if you rotate 180°) f(-x) = -f(x) multiply f(x) by -1 to get –f(x) then simplify compare to f(-x), if same then odd If (a, b) is on the graph of f(x) then (-a, -b) will also be on the graph.

Continuity Continuous function: The graph can be drawn from -∞ to +∞ without lifting the pen/pencil and there are no sharp turns. Examples of continuous functions: (constant, linear, quadratic, other polynomials, some rational) Examples of non-continuous functions: (absolute value, piecewise, many rational) Types of discontinuity Removable discontinuity: hole in the graph (division by zero) Infinite discontinuity: output values shoot up to positive infinity or down to negative infinity (also division be zero) Jump discontinuity: There is a jump up or down from one part of the graph to another (piecewise functions)

Graphing piecewise functions on a TI-83+ calculator y = (Rule 1)(Condition 1) + (Rule 2)(Condition 2) + … Inequality symbols are found in the TEST menu which can be reached by pressing 2nd MATH May be useful to switch calculator from connected to dot mode.

Intermediate Value Theorem If f(x) is a continuous function and a < b then f(x) is equal to every y-value between f(a) and f(b) at least once. This theorem is most useful when the signs of f(a) and f(b) are opposites. In this case, we know that the graph of the function crossed the x-axis at least once in the interval (a, b).

End Behavior What’s happening to f(x) values on the left end of the graph (i.e.: as x-values approach negative infinity)? are they going up (approaching positive infinity)? are they going down (approaching negative infinity)? are they leveling off (approaching a certain value)? What’s happening to f(x) values on the right end of the graph (i.e.: as x-values approach positive infinity)?

Increasing/Decreasing/Constant Intervals Increasing Interval: y-values are going up from Left to Right Decreasing Interval: y-values are going down from L. to R. Constant Interval: y-values remain the same from L. to R. x-values are always used to describe intervals

Relative Extrema Relative maximum: highest point in an interval. Relative minimum: lowest point in an interval. points are used to describe max and mins

Average Rate of Change Use the slope formula and the beginning and ending x-values for a given interval. Slope formula

Parent Graphs f(x) = c constant f(x) = x linear (identity) f(x) = x2 quadratic f(x) = x3 cubic f(x) = | x | absolute value f(x) = square root greatest integer f(x) = 1/x rational (reciprocal)

Transformations of a parent graph g(x) = a f(bx + c) + d Sequence of Transformations: c, b, a, d a Vertical expansion if |a| > 1 Vertical compression if |a| < 1 Vertical reflection if a < 0 b Horizontal compression if |b| > 1 Horizontal expansion if |b| < 1 Horizontal reflection if b < 0 c Horizontal shift: left if c > 0, right if c < 0 d Vertical shift: up if d > 0, down if d < 0 Generalizations: Inside affect horizontal opposite of what you expect Outside affects vertical the way you expect Multiply affects expansion/compression Mult by neg affects reflection Add/Subtract affects shift

Function Operations (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f ÷ g)(x) = f(x) ÷ g(x); g(x) ≠ 0 Recall that f(x) and g(x) are the outputs or y-values for a given x-value. Therefore (f + g)(x) means to add the two y-values associated with the given x-value. Function Operations: Combine two outputs.

Composition of Functions (f º g)(x) = f( g( x ) ) Input x into g(x) find the output for g(x) input the result into f(x) find the output for f(x) The output of the inner function becomes the input of the outer function.

Inverse Functions The inputs and the outputs switch roles. the input of f(x) becomes the output of f-1(x) the output of f(x) becomes the input of f-1(x) If (a,b) is on the graph of f(x), then (b,a) will be on the graph of f-1(x) Horizontal Line Test If it works, then the function is a one-to-one function and its inverse will be a function. If it does not work, then the function is not a one-to-one function and the inverse will not be a function.

Verifying Inverse Functions f and g are inverses if and only if f(g(x)) = g(f(x)) = x Find the composite of f(g(x)). If not equal to x, then f and g are not inverses If equal to x, then continue by finding composite of g(f(x)) If equal to x, then f and g are inverses.

Finding Inverses Algebraically Replace f(x) with y. Switch x and y. Solve for y. Replace y with f-1(x).