In Problems 11-20, for the given functions f and g, find: (a) (f ∘g)(4) (b) (g ∘f)(2) (c) (f ∘f)

In Problems 11-20, for the given functions f and g, find:...

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Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the results of another, expressed as (f ? g)(x) = f(g(x)). This concept emphasizes the order of operations where the inner function is evaluated first, and then the result is passed to the outer function, making it fundamental in understanding how functions interact.

Evaluating Functions

Evaluating functions is the process of substituting a specific value into a function's expression to determine its output. This requires careful arithmetic and attention to the function's definition, and it is especially important when working with compositions where multiple layers of function evaluations are involved.

Inner and Outer Functions

In the composition of functions, the inner function is the one that is applied first, and its output becomes the input for the outer function. Recognizing which function plays the role of the inner or outer component is critical, as it affects the outcome of the composite function and helps in systematically solving problems involving compositions.

Transcript

00:01 This problem gives us the f of x function of 2x and the g of x function of 3x squared plus 1, and we're asked for different function compositions evaluated at given input values.

00:11 And to be able to figure this out, we need to remember what this notation means.

00:14 When we're asked f of g of x, whether we have an actual value for x or not, what that's happening, or what's happening to that, and what we're supposed to do is take the g of x function and evaluate the outer f of x function with that value.

00:28 So just as an example for a, when we're asked f of g of 4, that means we're taking g of 4, whatever the result is, and we are inputting it into the f of x function.

00:40 So we're evaluating f of g of 4.

00:42 And to do that, we need to know what g of 4 is first.

00:45 So that would be 3 times 4 squared plus 1.

00:49 That's 16 times 3, which is 48 plus 1, and that gives us 49.

00:54 So that means this problem has now turned into f of 49 instead of f of g of 4.

01:01 And now we'll evaluate f of x, which would be 2 times 49, and that gives us 98.

01:07 So that tells us the result of a would be equal to 98.

01:11 For b, we will do the same thing, except this time we're finding f of 2 first.

01:15 So we're evaluating g at f of 2.

01:18 F of 2 would be 2 times 2, which is 4...