Domain Of Composite Functions: A Step-by-Step Guide

In this article, we'll explore how to find the domain of a composite function. Specifically, we'll tackle the problem: If $a(x) = 3x + 1$ and $b(x) = \sqrt{x - 4}$, what is the domain of $(b \circ a)(x)$?

Understanding Composite Functions

Before we dive into the problem, let's clarify what a composite function is. A composite function is a function that is formed by substituting one function into another. In our case, $(b \circ a)(x)$ means we are plugging the function $a(x)$ into the function $b(x)$. Mathematically, this is written as $b(a(x))$.

Key Idea: The domain of a composite function is not always obvious and requires careful consideration of the domains of both the inner and outer functions.

Step-by-Step Solution

Let's break down the process of finding the domain of $(b \circ a)(x)$.

1. Find the Composite Function $(b \circ a)(x)$

First, we need to find the expression for $(b \circ a)(x)$. We do this by substituting $a(x)$ into $b(x)$:

$b(a(x)) = b(3x + 1) = \sqrt{(3x + 1) - 4} = \sqrt{3x - 3}$

So, $(b \circ a)(x) = \sqrt{3x - 3}$.

2. Determine the Domain of the Composite Function

Now, we need to find the domain of $\sqrt{3x - 3}$. Remember that the domain of a square root function is all real numbers for which the expression inside the square root is non-negative (i.e., greater than or equal to 0). This is because we cannot take the square root of a negative number and get a real number result.

So, we need to solve the inequality:

$3x - 3 \geq 0$

Add 3 to both sides:

$3x \geq 3$

Divide both sides by 3:

$x \geq 1$

This means the domain of $(b \circ a)(x)$ is all real numbers $x$ such that $x \geq 1$.

3. Consider the Domain of the Inner Function $a(x)$

It's crucial to also consider the domain of the inner function, $a(x)$. In this case, $a(x) = 3x + 1$. This is a linear function, and linear functions have a domain of all real numbers. So, the domain of $a(x)$ is $(-\infty, \infty)$.

4. Consider the Domain of the Outer Function $b(x)$

We also need to consider the domain of the outer function, $b(x)$. In this case, $b(x) = \sqrt{x-4}$. The domain of $b(x)$ requires $x-4 \geq 0$, which means $x \geq 4$. So the domain of $b(x)$ is $[4, \infty)$.

5. Combine the Domain Restrictions

The domain of the composite function $(b \circ a)(x)$ must satisfy the restrictions imposed by both the inner and outer functions. We found that the inner function $a(x)$ has a domain of all real numbers. For the outer function $b(x)$, we require that the input to $b(x)$, which is $a(x) = 3x + 1$, must be greater than or equal to 4.

So, we have the condition:

$3x + 1 \geq 4$

Subtract 1 from both sides:

$3x \geq 3$

Divide both sides by 3:

$x \geq 1$

Therefore, the domain of $(b \circ a)(x)$ is $[1, \infty)$.

Final Answer

The domain of $(b \circ a)(x)$ is $[1, \infty)$. Therefore, the correct answer is C. $[1, \infty)$.

Why This Matters: Domain Restrictions Explained

Alright, so why do we even bother with domain restrictions? Think of functions like machines. You feed them a number (an input), and they spit out another number (an output). But some machines are picky! They can't handle just any input.

Square Root Functions: These are the classic example. You can't take the square root of a negative number (at least not and get a real number back). So, the stuff inside the square root must be zero or positive. That's a domain restriction!

Rational Functions (Fractions): You can't divide by zero. So, if you have a fraction with 'x' in the denominator, you need to make sure the denominator never equals zero. Again, domain restriction!

Logarithmic Functions: Logarithms are only defined for positive numbers. So, the argument of a logarithm (the thing you're taking the log of) must be positive. Yep, another domain restriction!

When we're dealing with composite functions, things get a little trickier because we need to consider the domains of all the functions involved. It's like a chain reaction: if the inner function spits out something that the outer function can't handle, the whole thing breaks down!

Common Mistakes to Avoid

Finding the domain of composite functions can be tricky, so let's look at some common pitfalls and how to avoid them.

Forgetting the Inner Function's Domain

A very common mistake is to only consider the domain of the final composite function without considering the domain of the inner function. Remember, the input to the outer function must be in the domain of the outer function. This means that the output of the inner function must be a valid input for the outer function.

Always check the domain of the inner function first. If there are any restrictions on the input of the inner function, those restrictions also apply to the composite function.

Incorrectly Solving Inequalities

When finding the domain of functions involving square roots or other restrictions, you'll often need to solve inequalities. Make sure you're comfortable with the rules for solving inequalities, such as when to flip the inequality sign (when multiplying or dividing by a negative number).

Double-check your work when solving inequalities to avoid making careless errors.

Not Considering All Restrictions

Sometimes, a composite function may have multiple restrictions on its domain. For example, it might involve both a square root and a rational function. In this case, you need to consider all the restrictions and find the intersection of the domains.

Make sure you identify all the potential domain restrictions and combine them correctly.

Overlooking the Obvious

Sometimes, the domain restriction is simply that x cannot be a certain value. For example, if we had a function like f(x) = 1/(x-2), then x cannot be 2, or the function is undefined. This needs to be considered when finding the domain of a composite function.

Practice Problems

To solidify your understanding, try these practice problems:

1. If $f(x) = \frac{1}{x}$ and $g(x) = x + 2$, find the domain of $(f \circ g)(x)$.
2. If $f(x) = \sqrt{x + 3}$ and $g(x) = x^2 - 4$, find the domain of $(f \circ g)(x)$.
3. If $f(x) = \frac{1}{x-1}$ and $g(x) = \sqrt{x}$, find the domain of $(g \circ f)(x)$.

Conclusion

Finding the domain of composite functions requires a systematic approach. By carefully considering the domains of both the inner and outer functions, you can accurately determine the domain of the composite function. Remember to avoid common mistakes and practice regularly to master this skill. Keep these tips in mind, and you'll be navigating composite function domains like a pro! And always remember to double-check your work. Good luck!