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Understanding Function Notation in Algebra

Learn how a function is a process matching inputs to outputs, and delve into algebraic formulas for functions. Explore examples and understand the concept of independent and dependent variables. Discover implied domains and real-world applications.

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Understanding Function Notation in Algebra

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  1. Section 1.4 Function Notation

  2. Function as a Process • The domain of a function is a set of inputs and • The range is a set of outputs • A function f is a process by which each input x is matched with only one output y • y = f(x) is the output which results by applying the process f to the input x

  3. Independent and Dependent Variables • The value of y is completely dependent on the choice of x • x is often called the independent variable or argument of f • y is often called the dependent variable

  4. Algebraic Formula of Function • The process of a function f is usually described using an algebraic formula • Example: a function f takes a real number and performs the following two steps • multiply by 3 • add 4 • If 5 is our input, in step 1 we multiply by 3 to get (5)(3) = 15. In step 2, we add 4 to our result from step 1 which yields 15 + 4 = 19 • Using function notation, we would write f(5) = 19 to indicate that the result of applying the process f to the input 5 gives the output 19. • In general, if we use x for the input, applying step 1 produces 3x. Following with step 2 produces f(x) = 3x + 4 as our final output.

  5. Example • Suppose a function g is described by applying the following steps, in sequence • add 4 • multiply by 3 • Determine g(5) and find an expression for g(x)

  6. Example • For f(x) = -x2 + 3x + 4, find and simplify • f(-1), f(0), f(2) • f(2x), 2 f(x) • f(x + 2), f(x) + 2, f(x) + f(2)

  7. Implied Domain • Suppose we wish to find r(3) for r(x) =2x/(x2 – 9) • The number 3 is not an allowable input to the function r; in other words, 3 is not in the domain of r • When a formula for a function is given, we assume the function is valid for all real numbers which make arithmetic sense when substituted into the formula. • This set of numbers is often called the implied domain of the function.

  8. Example • Find the domain of the following functions

  9. Example • This example shows how a function can be used to model real-world phenomena • The height h in feet of a model rocket above the ground t seconds after lift off is given by • Find and interpret h(10) and h(60) • This type of function is called a piecewise function

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