Problem 63 - Precalculus Essentials

Question

Let $$ \begin{array}{l} f(x)=2 x-5 \\ g(x)=4 x-1 \\ h(x)=x^{2}+x+2 \end{array} $$ Evaluate the indicated function without finding an equation for the function. $$g(f[h(1)])$$

Step-by-Step Solution

Answer

The value of \(g(f[h(1)])\) is 11.

Step 1: Evaluate the innermost function

Start by evaluating the function \(h(1)\). Substituting \(x = 1\) into \(h(x) = x^{2} + x + 2\) yields \(h(1) = 1^{2} + 1 + 2 = 4\).

Step 2: Evaluate the next outer function

The result \(h(1) = 4\) is used as the input for the next function \(f(x)\). Hence, we substitute \(x = 4\) into \(f(x) = 2x - 5\) to get \(f(4) = 2*4 - 5 = 3\).

Step 3: Evaluate the outermost function

Finally, the result \(f(4) = 3\) is used as the input for the function \(g(x)\). Substitute \(x = 3\) into \(g(x) = 4x - 1\) to get \(g(3) = 4*3 - 1 = 11\).

Key Concepts

Function Evaluation

Function evaluation involves substituting a given input value into a function to find the output. It’s like asking the function a question. In our task, we find the function outputs for specific inputs step by step.

Let’s break it down with a simple example. Consider the function \( f(x) = 2x - 5 \). If we want to find \( f(4) \), we substitute 4 for \( x \) in the function:

• Plug in 4: \( f(4) = 2\times4 - 5 \).
• Calculate: \( 2\times4 = 8 \).
• Subtract 5: \( 8 - 5 = 3 \).

So, \( f(4) = 3 \). This simple process is the essence of function evaluation—a key step in working with composite functions.

Nested Functions

Nested functions occur when you have functions within functions, like a set of Russian dolls. Each function feeds into the next. For composite functions, this nesting needs careful evaluation from the innermost to the outermost function.

In our exercise, we evaluated \( g(f(h(1))) \). Here:

• \( h(1) \) is the innermost function and was evaluated first to get 4.
• Next, \( f(4) \) took the result of the first function, leading to 3.
• Finally, \( g(3) \) used the second result to get the final output, 11.

It’s like a step-by-step journey, following the path from inner operations outward, ensuring each function’s output becomes the next’s input. This sequential order is crucial for accurate results.

Function Operations

Function operations involve combining functions through addition, subtraction, multiplication, or division, and through composition—which is different from basic operations. Composition connects multiple functions into a single operation sequence.

In our example, we used composition by applying one function to the output of another, forming a chain: \( g(f(h(x))) \). This isn't just about arithmetic operations but about plugging one function’s result into another as an input.

To handle composite functions effectively, remember:

• Identify each individual function and their domains.
• Carefully track the results at each stage of composition.
• Approach composition sequentially—inside out.

Function operations, particularly composition, offer flexibility in mathematics, allowing the construction of complex models from simpler functions.