Construct the following composite functions and specify the domain of each. (a) f ∘f(x) (b) f ∘g

Construct the following composite functions and specify the...

Key Concepts

Composite Functions

A composite function is formed when you apply one function to the result of another function. This concept is fundamental in algebra and calculus, as it allows for building more complex functions from simpler ones, and requires careful consideration of the domains of the individual functions to ensure that the composition is meaningful.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. When working with composite functions, it is important to consider not only the domain restrictions of the outer function, but also ensure that the values produced by the inner function fall within the domain of the outer function.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. They are a common subject in algebra and calculus, and understanding their properties, particularly how to determine where they are undefined (typically when the denominator is zero), is key when composing and analyzing such functions.

Sign Function

The sign function, often denoted as sgn(x), is a simple function that extracts the sign of a real number. It returns values that indicate whether a number is positive, negative, or zero. This function is commonly used in various areas of mathematics and can introduce unique domain considerations when used in compositions with other functions.

Transcript

00:01 For this problem, we are asked to construct each one of the composite functions shown here, using f of x equals x plus 1 over x minus 1, and g of x equals the sine sgn function of x, which is just going to be negative 1 when x is less than 0.

00:16 It's going to be positive 1 when x is greater than 0.

00:19 So i'll note first that the domain of f of x, well, we have this division by x minus 1 happening here.

00:30 So the domain is going to be x in the real numbers except for zero, or not zero, excuse me, except for one.

00:38 If we have x equals one, then we'd have division by zero.

00:41 We can't have that.

00:43 We also would have that the domain of g of x is going to be the entirety of the real numbers, except for x equals zero.

00:54 Having that, we can start with part a there, f of f of x.

01:00 So that composite function, f of f of x, it's going to be the result simply of substituting in f of x for x in each one of the places that x appears in the original function.

01:14 So we would have x plus 1 over x minus 1 plus 1 divided by x, oopsie, daisy, x plus 1 over x minus 1 minus 1.

01:29 So we'd have then, we can say, simplify this by multiplying both numerator and denominator through, or not, excuse me, not multiplying, but rather putting both numerator and denominator over the common factor of x minus 1.

01:46 So we'd have up top x plus 1 plus x minus 1.

01:50 Then in the denominator, we'd have x plus 1 minus x plus 1 divided by x minus 1.

01:58 Now since we are dividing by a fraction that's going to be the same thing as multiplying.

02:03 So we're going to be the same thing as multiplying.

02:03 We can note that we'd have the x minus ones divide out.

02:07 Then up top, we'd have x plus 1 minus 1, so we'd have 2x.

02:12 And because of the nature of that division, we would have 2x up top, and then in the denominator, we'd have x minus x, so 0 plus 2.

02:22 So we'd have 2x over 2, giving us x as the most simplified form.

02:26 But we need to note that while x is defined for the entirety of the real numbers, we do have the restriction that we have to have something that we can only have the x values that generate values of f x that are in the domain of f of x.

02:42 So specifically here the domain is going to be the set of x in the real numbers such that x does not equal one.

02:55 Then moving along to part b, f of g of x, well that is going to be the result of inserting g of x...

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