For each pair of functions, determine a) (f ∘g)(x), b)(f...

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Key Concepts

Function Composition

Function composition is the process of applying one function to the result of another. If you have two functions f and g, the composite function (f ? g)(x) is defined as f(g(x)). This concept is essential in mathematics as it allows for combining simpler functions to build more complex ones and analyze their behavior.

Domain Restrictions

When composing functions, it is important to consider the domain restrictions of each function. Every function has a domain—the set of input values for which it is defined—and when composing two functions, the domain of the composite function consists of all values that satisfy the requirements of both the inner and outer functions.

Evaluation of Composite Functions

Evaluating composite functions involves substituting the output of the inner function into the outer function. This process typically follows the order: first, compute g(x) for the given x, and then apply the function f to the result of g(x) to obtain (f ? g)(x). Similarly, for (g ? f)(x), you first compute f(x) and then substitute that into g.

Transcript

00:01 So we're given two functions.

00:02 F of x is equal to 4x squared minus 7x minus 1, and g of x is equal to x squared plus 3x minus 6.

00:08 And for part a, we have to evaluate the expression f plus g of x.

00:12 And once we use function algebra, we realize this is simply asking us to add together the functions f of x plus g of x.

00:21 So now we can use what we have here, form our expression, 4x squared minus 7x minus 1 is equal to fx, plus x squared minus 3x minus 6.

00:40 Now we have this, we can try solving out x4x squared minus 7x minus 1 plus x squared minus 3x minus 6.

00:54 And now we have this, so 4x squared plus x squared is 5x squared.

01:00 5x squared.

01:02 And negative 7x minus 3x is negative 10x, negative 1 minus negative 6 is equal to negative 7.

01:12 So we have the answer to part a, which is 5x squared minus 10x minus 7.

01:19 Now for part b, it's asking us to evaluate f plus g, which we found right over here.

01:25 But using 5 as x.

01:27 So it wants us to substitute 5 into this equation.

01:31 And so now we get 5 times 5 squared minus 10 times 5 minus 7.

01:40 So 5 squared is 25, 25 times 5 is 125 minus 50 minus 7.

01:49 Now we have, simplify this and we get 75 minus 7, which is equal, to 68.

02:04 68 is the answer to part e.

02:08 Now for part c, we're asked to evaluate the expression f minus g of x...

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