Day 3 examples
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Composite functions refer to combining two functions where the output of one acts as the input of the other. The notation used is (f ◦ g)(x) which reads "f composed with g of x", where the inner function g(x) is evaluated first, then substituted into the outer function f(x). Examples show how to evaluate composite functions by first substituting the inner function and then evaluating the outer function.
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2.4 operations on functions
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Composite functions refer to combining two functions, where the output of one function is used as the input of the other. The notation used is (f ◦ g)(x) = f(g(x)), where the inner function g(x) is evaluated first, and then substituted into the outer function f(x). Some examples are given of evaluating composite functions using function tables or graphs.
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Day 3 examples
- 1. Composite Functions Refers to the combining of two functions f(x) and g(x) where the output of one function is used as the input of the other function. Recall, f(x) = 2x - 1 find f(3) x=3 The notation used for compostion is Inner brackets are done first. (f ◦ g)(x) = f(g(x)) First substitute into g, then into f. Reads "f composed with g of x" or "f of g of x"
- 2. Ex 1) If f(x) = x + 1 and g(x) = 2x, find (f ◦ g)(x). Steps: 1) Write the expression for the function of g (2x) in the g(x) 'spot' in the composition 2) Now substitute this expression (2x) into function f in the x 'spot' 3) Simplify (if necessary)
- 3. Ex 2) Given f(x) = 5x and g(x) = x2 + 1, find a) (f ◦ g)(x) b) (g ◦ f)(x) Notice that (f ◦ g)(x) and (g ◦ f)(x) do not necessarily have the same answer.
- 4. Ex 3) Given f(x) = x2 and g(x) = x + 3, find a) f(g(x)) b) g(f(x)) c) f(f(x)) d) g(g(x))
- 5. Ex 4) If f(x) = 4x, g(x) = x + 6, and h(x) = x2, find a) f(g(3)) b) g(h(-2)) c) h(h(2)) Two options possible