Functions can be combined in two ways:
Composition feeds one function’s output into another, while operations on functions combine function values at the same input.
This week, we’ll practice both with three functions. Building on last week’s work with domains and ranges, we’ll use those ideas to check the resulting expressions.
The functions:
Compose From the Inside Out
In f composed of g, apply g first, then use its outputs as the input to f:
To find the domain, we start by saying the radicand must be positive:
But notice for any real number x, the square or the left hand side will always be positive:
So the interval is:
Thus:
Now apply h to the output of f composed of g:
The denominator must not be zero. Therefore:
So:
Combine Function Values With Operations
For the product of f and g:
If we divide by h, we simply divide the product by the function of h:
For this new functions domain, we must account for the domains of f, g, and h. We also have an added restriction where h of x must not equal zero. Since h of x is never zero where it is defined, we have:
The point x equals two stays excluded even though the simplified expression has a value there. The original h of two is undefined.
Finding the Range of f Composed of g
The completed-square form of the composition makes its outputs visible. We have:
So the range must be a subset of:
For the reverse inclusion:
Solving for x, we have:
And since we have:
The reverse inclusion holds:
Hence:
Finding the Range of h Composed of f Composed of g
We have two cases for the outputs:
Evaluating the first case:
Evaluating the second case:
If we take the reciprocal, we get:
Combining the two cases, the range of this composition is a subset of:
Now solve the reverse inclusion:
Using the first condition for y:
Now we just need to make sure the values of x are never two or three, so we find an upper bound:
So we have an interval where x is never two or three:
In this case the y condition is in the domain of x. Now we verify the second case:
This confirms two and three are not in the domain and hence the unionized interval for y is a subset of the range. We conclude this is indeed the range:
Use the Composition to Solve an Equation
To solve:
Write out the composition:
Both values are real and belong to the domain of f composed of g which is all real numbers. For each, the radicand is nine, so both produce the output three.
Takeaway:
Composition passes an output into another function. Operations on functions combine function values at one input. Domain and range checks help confirm that each resulting rule behaves as expected.
Until next week,
Peace.✌️