Evaluating Composite Functions F(g(x)) And G(f(x))

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Let's delve into the world of composite functions with a focus on understanding how to evaluate them. Composite functions involve combining two or more functions, where the output of one function becomes the input of another. We'll explore this concept through a detailed example and then extend it to a broader discussion. To clarify how the function composition will work we need to consider these functions:

f(x)=16x2g(x)=14x\begin{array}{l} f(x)=16 x^2 \\ g(x)=\frac{1}{4} \sqrt{x} \end{array}

For $x \geq 0$, the value of $f(g(x))$ needs to be evaluated and then we will move to next function composition that is $g(f(x))$.

Evaluating $f(g(x))$

The core concept of composite functions is the sequential application of functions. When we encounter $f(g(x))$, it signifies that we first apply the function $g$ to the input $x$, and then we take the output of $g(x)$ and use it as the input for the function $f$.

In simpler terms, we're plugging the entire function $g(x)$ into the function $f(x)$ wherever we see $x$. This might sound a bit abstract, so let's break it down with our specific functions.

  1. Start with the outer function: We have $f(x) = 16x^2$. This is our primary function, the one we'll be working with in the end.

  2. Identify the inner function: The inner function is $g(x) = \frac{1}{4}\sqrt{x}$. This function will be evaluated first.

  3. Substitute: Now, the crucial step. We replace every instance of $x$ in $f(x)$ with the entire expression of $g(x)$. So, we get:

    f(g(x))=16(14x)2f(g(x)) = 16 \left(\frac{1}{4}\sqrt{x}\right)^2

    Notice how the whole $g(x)$ expression, $ rac{1}{4}\sqrt{x}$, is now inside the parentheses where $x$ used to be in $f(x)$.

  4. Simplify: The final step is to simplify the expression. This involves following the order of operations and algebraic rules.

    • First, square the term inside the parentheses:

      (14x)2=116x\left(\frac{1}{4}\sqrt{x}\right)^2 = \frac{1}{16}x

    • Now, substitute this back into our expression:

      f(g(x))=16â‹…116xf(g(x)) = 16 \cdot \frac{1}{16}x

    • Finally, multiply:

      f(g(x))=xf(g(x)) = x

Therefore, the value of $f(g(x))$ for $x \geq 0$ is simply $x$. This might seem surprising, but it highlights how functions can interact in interesting ways when combined.

Evaluating $g(f(x))$

Now, let's reverse the order and find $g(f(x))$. This will give us a different perspective on composite functions and emphasize that the order in which we apply the functions matters significantly. This will often lead to different results.

Following the same logic as before, $g(f(x))$ means we first apply the function $f$ to $x$, and then use the result as the input for the function $g$.

  1. Start with the outer function: In this case, our outer function is $g(x) = \frac{1}{4}\sqrt{x}$.

  2. Identify the inner function: The inner function is now $f(x) = 16x^2$.

  3. Substitute: Replace every $x$ in $g(x)$ with the entire expression of $f(x)$. This gives us:

    g(f(x))=1416x2g(f(x)) = \frac{1}{4}\sqrt{16x^2}

    Again, notice how the whole $f(x)$ expression, $16x^2$, is placed inside the square root where $x$ used to be in $g(x)$.

  4. Simplify: Let's simplify this expression step by step:

    • First, simplify the square root. Since we're given that $x \geq 0$, we can take the positive square root:

      16x2=4x\sqrt{16x^2} = 4x

    • Substitute this back into our expression:

      g(f(x))=14(4x)g(f(x)) = \frac{1}{4}(4x)

    • Finally, multiply:

      g(f(x))=xg(f(x)) = x

Thus, the value of $g(f(x))$ for $x \geq 0$ is also $x$. In this particular case, we see that $f(g(x))$ and $g(f(x))$ both simplify to $x$. This outcome indicates a special relationship between the functions $f$ and $g$, which we'll explore further.

Inverse Functions and Composition

The fact that both $f(g(x))$ and $g(f(x))$ resulted in $x$ suggests that $f(x)$ and $g(x)$ are inverse functions of each other. This is a crucial concept in mathematics.

Inverse functions are functions that "undo" each other. If you apply a function and then apply its inverse, you end up back where you started. More formally, if $f$ and $g$ are inverses, then:

  • f(g(x)) = x$ for all $x$ in the domain of $g

  • g(f(x)) = x$ for all $x$ in the domain of $f

Our example perfectly illustrates this. When we composed $f$ with $g$ or $g$ with $f$, the final result was always the original input $x$. This is the defining characteristic of inverse functions.

How to Find Inverse Functions

To confirm that $f(x) = 16x^2$ (for $x \geq 0$) and $g(x) = \frac{1}{4}\sqrt{x}$ are inverses, we can also follow the standard procedure for finding inverse functions:

  1. Replace $f(x)$ with $y$:

    y=16x2y = 16x^2

  2. Swap $x$ and $y$:

    x=16y2x = 16y^2

  3. Solve for $y$:

    • Divide by 16:

      x16=y2\frac{x}{16} = y^2

    • Take the square root (remembering $y \geq 0$ since the range of an inverse function becomes the domain of the original function):

      y=x16=14xy = \sqrt{\frac{x}{16}} = \frac{1}{4}\sqrt{x}

  4. **Replace $y$ with $f^{-1}(x)$

    f−1(x)=14xf^{-1}(x) = \frac{1}{4}\sqrt{x}

This result confirms that the inverse of $f(x)$ is indeed $g(x)$, reinforcing our observation about their inverse relationship based on the composite functions.

Domains and Ranges of Composite Functions

When working with composite functions, it's essential to consider the domains and ranges of the individual functions and how they affect the composite function.

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values).

In our example:

  • For $f(x) = 16x^2$, the domain is all real numbers, but since we are given $x \geq 0$, we're restricting the domain to non-negative numbers. The range is also non-negative real numbers.
  • For $g(x) = \frac{1}{4}\sqrt{x}$, the domain is $x \geq 0$ (because we can't take the square root of a negative number), and the range is also non-negative real numbers.

When we form the composite function $f(g(x))$, the domain is restricted by the domain of the inner function, $g(x)$. In this case, $g(x)$ is only defined for $x \geq 0$, so $f(g(x))$ is also only defined for $x \geq 0$. Similarly, for $g(f(x))$, the domain is restricted by the domain of $f(x)$, which is $x \geq 0$ in our given condition.

The Importance of Order in Composition

We've seen that the order in which we compose functions matters. In general, $f(g(x))$ is not the same as $g(f(x))$. The order affects how the functions interact and the final result.

In our specific example, we found that both compositions resulted in $x$, which is a special case indicating inverse functions. However, in most cases, changing the order of composition will lead to different outcomes.

To illustrate this further, consider two simpler functions:

  • f(x)=x+1f(x) = x + 1

  • g(x)=2xg(x) = 2x

Let's find $f(g(x))$ and $g(f(x))$

  • f(g(x))=f(2x)=2x+1f(g(x)) = f(2x) = 2x + 1

  • g(f(x))=g(x+1)=2(x+1)=2x+2g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2

As you can see, $2x + 1$ is clearly different from $2x + 2$, demonstrating that the order of composition matters.

Applications of Composite Functions

Composite functions are not just abstract mathematical concepts; they have practical applications in various fields.

  • Modeling real-world processes: Composite functions can be used to model situations where one process depends on another. For example, the cost of manufacturing a product might depend on the number of units produced, and the number of units produced might depend on the number of employees working. A composite function could model the overall cost as a function of the number of employees.
  • Computer graphics: In computer graphics, transformations like scaling, rotation, and translation are often represented as functions. Combining these transformations involves composing the corresponding functions.
  • Calculus: Composite functions play a crucial role in calculus, particularly in the chain rule, which helps us find the derivative of a composite function.

Conclusion

Understanding composite functions is essential for a solid foundation in mathematics. By carefully evaluating the functions step-by-step, considering domains and ranges, and recognizing the significance of order, we can confidently work with composite functions and appreciate their applications in various contexts. The special case of inverse functions, where composition results in the original input, highlights a fundamental relationship between functions and their ability to "undo" each other.