Problem 90 - College Algebra with Modeling and Visualization

Question

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=5 \sqrt{x-1} $$

Step-by-Step Solution

Answer

Functions are \(f(x) = x - 1\) and \(g(x) = 5\sqrt{x}\).

Step 1: Understand Function Composition

The given function is in the form of composition, where \(h(x) = (g \circ f)(x) = g(f(x))\). Our task is to find functions \(f(x)\) and \(g(x)\) such that their composition gives the desired function \(h(x) = 5\sqrt{x-1}\).

Step 2: Identify Inside Function

Recognize the expression under the square root as a hint to choose \(f(x)\). A common choice is the expression inside the outer function. In this case, let \(f(x) = x - 1\). This extracts the expression inside the square root.

Step 3: Identify Outside Function

Now identify the expression that operates on the result of \(f(x)\). Since \(g(x)\) must take the output of \(f(x)\) and transform it into \(5\sqrt{f(x)}\), let \(g(x) = 5\sqrt{x}\). This ensures that \(g(f(x)) = 5\sqrt{f(x)} = 5\sqrt{x-1}\).

Step 4: Verify Composition

Verify that the composition \(g(f(x))\) results in \(h(x)\). With \(f(x) = x - 1\) and \(g(x) = 5\sqrt{x}\), compute \(g(f(x)) = g(x-1) = 5\sqrt{x-1}\), which matches \(h(x)\).

Key Concepts

Identifying Functions

Recognizing what makes something a function is crucial when working with function compositions. A function is essentially a rule that assigns every input exactly one output. This means there's a predictable and consistent relationship between inputs and their corresponding outputs.

When you encounter a function like \( h(x) = 5\sqrt{x-1} \), you're dealing with a transformation of inputs into a specific form of outputs. You'll often need to "break down" these transformations into simpler components. That's the crux of identifying functions for composition: spotting the parts that make up the whole function. You divide \( h(x) \) into two simpler functions, \( f(x) \) and \( g(x) \), which when combined recreate \( h(x) \).

In practice:

• Examine the given expression \( h(x) \).
• Determine the "inside" function \( f(x) \) which is most often a part of the main expression.
• Identify the "outside" function \( g(x) \) which transforms the result of \( f(x) \).

This clear breakdown into parts aids in understanding more complex functions deeply and intuitively.

Square Root Function

The square root function is essential in mathematics, often seen in forms where inputs are transformed by extracting their square root. It's defined as \( g(x) = \sqrt{x} \) and is known for creating a curve that rises slowly as \( x \) increases.

The uniqueness of the square root function lies in its property to only output non-negative results when dealing with real numbers. This means the input also needs to be non-negative, thus creating a constraint in function composition.

When dealing with a composed function like \( h(x) = 5\sqrt{x-1} \), here:

• We've used the square root function as part of our outer function \( g(x) = 5\sqrt{x} \).
• Within this context, \( g(x) \) scales the output by 5.
• The transformation applied through \( g(x) \) highlights the flexibility and effect of the square root in creating a new output.

This scaling not only impacts the appearance of the graph but also modifies how inputs are perceived, providing a clear vision of the broader implications of the square root in functions.

Function Verification

Verification is a powerful step in ensuring the correctness of any function operation, including composition. It often involves evaluating if a recomposed function accurately reproduces the desired output.

To verify a composed function \( h(x) = g(f(x)) \), follow these steps:

• Identify given functions, here \( f(x) = x - 1 \) and \( g(x) = 5\sqrt{x} \).
• Substitute \( f(x) \) into \( g(x) \) to construct \( g(f(x)) = 5\sqrt{x-1} \).
• Compare \( g(f(x)) \) with the original function \( h(x) \).

If \( g(f(x)) \) matches \( h(x) \), then the functions \( f \) and \( g \) are correctly identified, which reinforces understanding and builds trust in your composition skills.

This approach to verification provides clarity and removes any ambiguity, making your study of functions precise and reliable.