Composite Functions: Find And Define Domains

Hey guys! Today, we're diving into the world of composite functions. We will be working with two functions, $f(x)=\sqrt{x-4}$ and $g(x)=5x$. Our mission is to find $(f \circ g)(x)$ and $(g \circ f)(x)$, and, super importantly, to determine their domains. Let's break it down step-by-step so you can nail this concept!

(a) Finding $(f \circ g)(x)$ and Its Domain

Step 1: What is $(f \circ g)(x)$?

First, let's clarify what $(f \circ g)(x)$ means. It represents the composition of function $f$ with function $g$, which is written as $f(g(x))$. In simpler terms, we're plugging the entire function $g(x)$ into function $f(x)$.

Step 2: Compute $(f \circ g)(x)$

Given $f(x) = \sqrt{x-4}$ and $g(x) = 5x$, we need to find $f(g(x))$.

So, we substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(5x) = \sqrt{5x - 4}$

Therefore, $(f \circ g)(x) = \sqrt{5x - 4}$.

Step 3: Determine the Domain of $(f \circ g)(x)$

Now, let's find the domain of $(f \circ g)(x) = \sqrt{5x - 4}$. Remember, the domain is the set of all possible input values (x-values) for which the function produces a real number. Since we have a square root, the expression inside the square root must be greater than or equal to zero.

So, we need to solve the inequality:

$5x - 4 \geq 0$

Add 4 to both sides:

$5x \geq 4$

Divide by 5:

$x \geq \frac{4}{5}$

Thus, the domain of $(f \circ g)(x)$ is all $x$ such that $x \geq \frac{4}{5}$. In interval notation, this is $\left[\frac{4}{5}, \infty\right)$.

In summary, $(f \circ g)(x) = \sqrt{5x - 4}$, and its domain is $\left[\frac{4}{5}, \infty\right)$.

Detailed Explanation of Domain

The domain of a function is a critical concept in mathematics. It specifies the set of input values for which the function is defined. When dealing with composite functions, determining the domain requires careful consideration of both the inner and outer functions. In this case, we started with $f(x) = \sqrt{x-4}$ and $g(x) = 5x$. When finding $(f \circ g)(x)$, we composed $f$ with $g$, resulting in $f(g(x)) = \sqrt{5x - 4}$. The square root function introduces a restriction because it is only defined for non-negative values. Therefore, we need to ensure that $5x - 4 \geq 0$. This inequality leads us to the condition $x \geq \frac{4}{5}$. This condition means that any value of $x$ less than $\frac{4}{5}$ would result in a negative value inside the square root, which is not defined in the realm of real numbers. Hence, the domain is all real numbers greater than or equal to $\frac{4}{5}$, represented in interval notation as $\left[\frac{4}{5}, \infty\right)$. Understanding the domain is crucial because it ensures that the function produces valid and meaningful results.

(b) Finding $(g \circ f)(x)$ and Its Domain

Step 1: What is $(g \circ f)(x)$?

Now, let's find $(g \circ f)(x)$, which means $g(f(x))$. This time, we're plugging the entire function $f(x)$ into function $g(x)$.

Step 2: Compute $(g \circ f)(x)$

Given $f(x) = \sqrt{x-4}$ and $g(x) = 5x$, we need to find $g(f(x))$.

So, we substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(\sqrt{x-4}) = 5(\sqrt{x-4})$

Therefore, $(g \circ f)(x) = 5\sqrt{x-4}$.

Step 3: Determine the Domain of $(g \circ f)(x)$

Now, let's find the domain of $(g \circ f)(x) = 5\sqrt{x-4}$. Again, we need to consider the square root. The expression inside the square root must be greater than or equal to zero.

So, we need to solve the inequality:

$x - 4 \geq 0$

Add 4 to both sides:

$x \geq 4$

Thus, the domain of $(g \circ f)(x)$ is all $x$ such that $x \geq 4$. In interval notation, this is $[4, \infty)$.

In summary, $(g \circ f)(x) = 5\sqrt{x-4}$, and its domain is $[4, \infty)$.

Comprehensive Domain Analysis

The domain determination for $(g \circ f)(x)$ involves analyzing the restrictions imposed by the function $f(x)$ because it is the inner function in this composition. Specifically, $f(x) = \sqrt{x-4}$ is only defined for $x$ values that satisfy $x - 4 \geq 0$. This inequality ensures that the value inside the square root is non-negative, which is a requirement for real-valued square root functions. Solving this inequality, we find that $x \geq 4$. Therefore, the domain of $f(x)$ is $[4, \infty)$. Since $f(x)$ is the input to $g(x)$, the domain of the composite function $(g \circ f)(x)$ is the same as the domain of $f(x)$, which is $[4, \infty)$. It is important to note that the function $g(x) = 5x$ itself has no domain restrictions, meaning it is defined for all real numbers. However, in the composite function, the domain is dictated by the restrictions on $f(x)$. Consequently, only $x$ values greater than or equal to 4 are valid inputs for $(g \circ f)(x)$, ensuring the function produces real and meaningful outputs.

Key Differences and Considerations

Order Matters

One key takeaway from this exercise is that the order of composition matters. $(f \circ g)(x)$ is generally not the same as $(g \circ f)(x)$. We saw that $(f \circ g)(x) = \sqrt{5x - 4}$ and $(g \circ f)(x) = 5\sqrt{x-4}$. These are different functions.

Domain Differences

Also, notice that the domains of $(f \circ g)(x)$ and $(g \circ f)(x)$ are different. The domain of $(f \circ g)(x)$ is $\left[\frac{4}{5}, \infty\right)$, while the domain of $(g \circ f)(x)$ is $[4, \infty)$. This difference arises from the specific restrictions imposed by the inner functions in each composition.

Importance of Domain

Always, always, always consider the domain when working with composite functions. It’s a critical step to ensure your function is well-defined and produces meaningful results.

Composition nuances

When composing functions, the domain of the inner function plays a crucial role in determining the domain of the composite function. In the case of $(f \circ g)(x)$, the domain is restricted by the need for $5x - 4$ to be non-negative under the square root, leading to $x \geq \frac{4}{5}$. Conversely, for $(g \circ f)(x)$, the domain is restricted by the need for $x - 4$ to be non-negative under the square root, leading to $x \geq 4$. These differences highlight that the order of composition matters significantly, not just in the resulting function but also in its domain. Failing to consider the domain can lead to incorrect or undefined results, making it an essential aspect of function analysis. Understanding these domain restrictions ensures that the composite functions are mathematically valid and yield meaningful outputs.

Conclusion

So, there you have it! We found that:

• $(f \circ g)(x) = \sqrt{5x - 4}$ with a domain of $\left[\frac{4}{5}, \infty\right)$.
• $(g \circ f)(x) = 5\sqrt{x-4}$ with a domain of $[4, \infty)$.

Remember, always pay close attention to the order of composition and the domains of the functions involved. Keep practicing, and you’ll become a composite function pro in no time! Happy calculating!