Problem 63 - Precalculus : Building Concepts and Connections

Question

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(h \circ f)(2)$$

Step-by-Step Solution

Answer

The value of \( (h \circ f)(2) \) is -18.

Step 1: Compute \(f(2)\)

We are given that \(f(x)=x^{2}+x\). Replacing \(x\) with \(2\), we perform the calculation to get \(f(2)=2^{2}+2 = 4+2 = 6.\)

Step 2: Compute \(h(f(2))\)

We computed and found that \(f(2) = 6\). Now replacing \(x\) in \(h(x)=-3x\) with 6, we get \(h(f(2))=h(6)=-3*6=-18\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function when given an input. Essentially, you substitute the input value into the function and compute the output. For example, given a function like \( f(x) = x^2 + x \), you can easily find the value of the function at a specific point, say \( x = 2 \), by plugging \( 2 \) into the function:

• Substitute: \( f(2) = 2^2 + 2 \)
• Calculate: \( f(2) = 4 + 2 \)
• Result: \( f(2) = 6 \)

Function evaluation is a critical first step in many calculus and algebra problems. It ensures you know precisely what a function does for specific input values. By mastering this concept, you can build a strong foundation for more complex operations such as finding limits, derivatives, or composing multiple functions.

Composite Functions

Composite functions are formed by applying one function to the results of another. In mathematical notation, if you have two functions, \( f \) and \( h \), the composite function is written as \((h \circ f)(x)\). It reads as "\(h\) of \(f\) of \(x\)."

• First, calculate \(f(x)\).
• Then, substitute this result into \(h(x)\).

For instance, say you want to find \((h \circ f)(2)\). You would: 1. Evaluate \(f(2)\), which we calculated as \(6\). 2. Next, evaluate \(h(6)\), giving us \(-18\).This step-by-step substitution might seem tricky at first, but once you think of composite functions as linking two simple evaluations together, they become much more intuitive. They are used frequently in calculus to solve more complex problems.

Precalculus Functions

Precalculus functions are the building blocks of advanced mathematics. Understanding how to manipulate these functions is crucial for anyone venturing into calculus or higher maths. Key precalculus functions include linear, quadratic, polynomial, exponential, and radical functions, among others.In our exercise, we interacted with a quadratic function \( f(x) = x^2 + x \) and a linear function \( h(x) = -3x \). Quadratic functions are notable for their parabolic shape and are written in the form \( ax^2 + bx + c \).

• Quadratic functions like \( f \) are typical precalculus functions offering insights into parabolas.
• Linear functions, such as \( h \), highlight the concept of a constant rate of change.

Understanding these functions arms you with the capability to tackle calculus problems, which frequently involve analyzing and composing functions to explore their properties and graphing behavior.