Problem 60 - 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. $$(g \circ f)(1)$$

Step-by-Step Solution

Answer

The value of \((g \circ f)(1)\) is \(\sqrt{2}\).

Step 1: Define the functions and the point

The functions given are \(f(x)=x^{2}+x\), \(g(x)=\sqrt{x}\), and the point given is \(x=1\). The aim is to evaluate \((g \circ f)(1)\) which means we will first calculate \(f(1)\) and then substitute that result into \(g(x)\).

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

By plugging \(x=1\) we get \(f(1)=1^{2}+1=1+1=2\).

Step 3: Evaluate \((g \circ f)(1)\)

Now we substitute the result of \(f(1) = 2\) into \(g(x)\). This gives us \(g(f(1)) = g(2) = \sqrt{2}\).

Step 4: Final Answer

Therefore, \((g \circ f)(1) = \sqrt{2}\)

Key Concepts

Function Composition

Function composition is the process of combining two or more functions in a specific order to create a new function. In other words, the output from one function becomes the input to another function. When dealing with composition, we use the notation \(g \circ f\) to mean 'g composed with f'. This tells us to first apply function \(f\) to our input, and then take that result and apply function \(g\) to it. Here’s an example to illustrate this concept:

If we have \(f(x) = x^2 + x\) and \(g(x) = \sqrt{x}\), and we want to find \(g \circ f\), we would first evaluate \(f\) at some value, say \(x = 1\). The result \(f(1)\) would then be used as the input for \(g\), leading to \(g(f(1))\). In the exercise provided, \(f(1) = 1^2 + 1 = 2\), and then \(g(f(1)) = g(2) = \sqrt{2}\). The ability to perform function composition is a fundamental concept in precalculus and integral to understanding higher-level mathematics.

Evaluating Functions

Evaluating functions is a crucial skill in mathematics that involves substituting a given value for the variable in a function's formula and simplifying the result. If you have a function \(f(x)\) and you're asked to find \(f(a)\), you would simply replace each occurrence of \(x\) in the formula with \(a\) and calculate the value.

For example, with \(f(x) = x^2 + x\), evaluating \(f(1)\) means we plug in \(1\) for every instance of \(x\) in the formula, giving us \(1^2 + 1 = 2\). It's important to work through the operations step by step to ensure accuracy, particularly when dealing with more complex functions or when compositions come into play. Evaluating functions is a common task in precalculus and sets the stage for understanding function behavior, graphing functions, and solving real-world problems using functions.

Precalculus

Precalculus is a course that prepares students for the rigors of calculus, and it encompasses a variety of topics including functions, composites of functions, trigonometry, and sequence and series.

In the context of our example, precalculus lays the groundwork for understanding how functions behave and how they can be manipulated. In precalculus, students learn to evaluate functions, understand the properties of different types of functions, and explore how these functions can be applied to solving problems. The function composition exercise from the textbook is an excellent demonstration of the kind of analytical thinking and procedural skills that are developed during a precalculus course. Proficiency in evaluating functions and understanding their compositions are stepping stones to more advanced concepts in calculus, such as derivatives and integrals, which are vital in fields such as physics, engineering, and economics.