Title 20 Functions.pdf ✅

1.2. Evaluation of Functions To evaluate a function, substitute the variable’s value and compute the resulting output. - Example: For the function f(x) = x 3 − 4x + 1, evaluate f (− 1 3 ). [SOLUTION] f (− 1 3 ) = (− 1 3 ) 3 − 4 (− 1 3 ) + 1 f (− 1 3 ) = 62 27 1.3. Domain and Range • Domain – The set of real numbers valid as input values for a function. • Range – It is the set of real numbers of all output values of a function. Shown in the table is the compilation of restrictions for common types of functions Function Form Domain Rational Functions f(x) g(x) g(x) ≠ 0 Irrational Functions √P(x) n f(x) ≥ 0 for even indices f(x) ∈ R for odd indices Quadratic Functions a(x − h) 2 + k All real numbers Exponential Functions a x ; a > 1 All real numbers Logarithmic Functions loga b a ∈ (0, 1] ∪ [1, ∞) And/or b > 0 - Example: Determine the domain of f(x) = √x − 3 x − 5 [SOLUTION] The restrictions are: x − 3 ≥ 0 → x ≥ 3 And x − 5 ≠ 0 → x ≠ 5 Therefore, the possible forms of the answer are x ≥ 3, x ≠ 5 x ∈ [3, 5) ∪ (5, ∞)
1.4. Parent Functions These fundamental functions that may be combined or transformed to produce other functions. This discussion shows the forms and graphs of the parent functions to give an idea of how to sketch the graph of a function. 1.4.1. Linear Functions • These functions, f(x) = ax + b, where a is the slope and b is the y-intercept. This form is also helpful in analytic geometry, known as the slope-intercept form. • Since it represents a line, at least 2 points are needed to graph it. Shown below is a sample graph of a linear function