Formulas for computing composite function

In summary, the conversation discusses two different ways to define the function h(x). One way defines h(x) as 0 for x ≤ 0 and x^2 for x > 0, while the other way defines h(x) as 0 for x < 0 and x^2 for x ≥ 0. Since the function values do not differ at x = 0, both definitions result in the same function.
  • #1
rxh140630
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11
Homework Statement
Let f and g be two functions defined as follows:

[itex] f(x) = \frac{x+|x|}{2}[/itex]

[itex] g(x) = \begin{cases}
x \text{ for x < 0} \\
x^2 \text{ for x ≥ 0}

\end{cases} [/itex]

Find a formula, or formulas, for computing the composite function h(x) = f[g(x)]
Relevant Equations
f ο g = f[g(x)]
h(x) = 0 for x ≤ 0
h(x) = x^2 for x>0

But my book says

h(x) = 0 for x<0
h(x) = x^2 for x≥0

Can my solution (the first one) work as well? Because the actual function value at x = 0 is zero. I feel like my solution is more elegant.
 
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  • #2
Yes, the both give the same result.
 
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  • #3
rxh140630 said:
h(x) = 0 for x ≤ 0
h(x) = x^2 for x>0

But my book says

h(x) = 0 for x<0
h(x) = x^2 for x≥0

These define the same function ##h##. To see this, you can ask at what points do the function values differ?
 
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  • #4
PeroK said:
These define the same function ##h##. To see this, you can ask at what points do the function values differ?

They do not differ because x^2 at x=0 = 0, if we choose to use the authors definition.

Since they do not differ then they must be the same.
 
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Related to Formulas for computing composite function

1. What is a composite function?

A composite function is a mathematical concept where one function is applied to the output of another function. It is denoted as f(g(x)), where g(x) is the inner function and f(x) is the outer function.

2. How do you compute a composite function?

To compute a composite function, you first substitute the input of the inner function into the variable of the outer function. Then, you simplify the resulting expression to get the final output.

3. What is the order of operations for computing a composite function?

The order of operations for computing a composite function is to first evaluate the inner function, then substitute the result into the outer function, and finally simplify the expression to get the final output.

4. Can you give an example of computing a composite function?

Sure, let's say we have the functions f(x) = x^2 and g(x) = 2x + 1. To compute f(g(x)), we first substitute g(x) into f(x), which gives us f(2x + 1) = (2x + 1)^2. Then, we simplify the expression to get the final output of 4x^2 + 4x + 1.

5. What are the common mistakes to avoid when computing a composite function?

Some common mistakes to avoid when computing a composite function include forgetting to substitute the inner function into the outer function, not following the correct order of operations, and making errors when simplifying the expression. It is important to double check your work and be careful with algebraic manipulations.

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