Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. \(f(x)=x^{2}+1, g(x)=\frac{1}{x},\) and \(h(x)=x+3\)

Step-by-Step Solution

Answer

\(f(g(h(x))) = \frac{1}{(x+3)^2} + 1\).

Step 1: Find the Innermost Composition

First, evaluate the innermost function, which is \(h(x)=x+3\). This means we need to substitute \(x+3\) into the function \(g(x)\) because \(g(h(x)) = g(x+3)\).

Step 2: Evaluate g(h(x))

Substitute the expression for \(h(x)\) into \(g(x)\). So, we have \(g(h(x)) = g(x+3) = \frac{1}{x+3}\).

Step 3: Substitute into f(x)

Now, take the result from Step 2 and substitute it into the function \(f(x)\). So, we have \(f(g(h(x))) = f\left( \frac{1}{x+3} \right)\).

Step 4: Evaluate f(g(h(x)))

Substitute \(\frac{1}{x+3}\) into \(f(x)\), which is \(f(x) = x^2 + 1\). This gives us \(f\left( \frac{1}{x+3} \right) = \left( \frac{1}{x+3} \right)^2 + 1\).

Step 5: Simplification

Simplify the expression \(\left( \frac{1}{x+3} \right)^2 + 1\) to get \( \frac{1}{(x+3)^2} + 1\). Thus, the simplified expression for \(f(g(h(x)))\) is \(\frac{1}{(x+3)^2} + 1\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols, often represented by letters such as \(x\) or \(y\), stand in for numbers and can represent various quantities.

Algebraic expressions can include operations such as addition, subtraction, multiplication, and division applied to these symbols and numbers. For example, in the function \(f(x) = x^2 + 1\), we have an algebraic expression involving a variable \(x\). In this context, algebra helps us to identify and simplify expressions, solve equations, and understand patterns and relationships.

When working with composite functions, you are combining algebraic operations from different functions into one single expression. This requires careful handling of variables and operations to ensure that the final expression is accurate and simplified.

Composite Functions

Composite functions involve combining two or more functions to form a new function. The notation \(f(g(h(x)))\) indicates that you first apply the function \(h(x)\), then \(g(x)\), and finally \(f(x)\).

Each step is a vital part of solving the problem.

• First, evaluate the innermost function, which is \(h(x)\). In this example, \(h(x) = x + 3\) shifts \(x\) by 3.
• Next, input the result into the next function, \(g(x)\), which modifies it with \(g(x) = \frac{1}{x}\).
• Finally, plug this result into \(f(x)\), giving \(f(g(h(x)))\).

Each function is applied sequentially, meaning you work from the inside out. This approach allows for a structured method of determining the overall function result, paving the way for simplification in the last step.

Step-by-Step Solution

Solving composite functions involves a clear and methodical approach. Here's a recap of the step-by-step solution to solve \(f(g(h(x)))\):

1. **Start with the innermost function:** Evaluate \(h(x) = x+3\). This transforms \(x\) by adding 3 to it.
2. **Substitute into the next function:** Calculate \(g(h(x)) = \frac{1}{x+3}\). This function takes the result from \(h(x)\) and transforms it by taking its reciprocal.
3. **Apply the outer function:** Plug the output from \(g(h(x))\) into \(f(x)\), giving \(f\left( \frac{1}{x+3} \right)\). Here, you square the expression and add 1, using \(f(x) = x^2 + 1\).
4. **Simplification:** Simplify the expression to \(\frac{1}{(x+3)^2} + 1\). You simplify by performing operations like squaring, reciprocals, and adding to achieve the final expression.

This systematic approach, from evaluating eachinner function to simplifying the expression, helps in understanding and solving problems involving composite functions effectively.