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Chapter 2: Problem 27

Find the rules for the composite functions \(f \circ g\) and \(g \circ f\). \(f(x)=\sqrt{x}+1 ; g(x)=x^{2}-1\)

Short Answer

Expert verified
The composite functions for the given functions \(f(x) = \sqrt{x} + 1\) and \(g(x) = x^{2} - 1\) are: \(f \circ g(x) = \sqrt{x^{2} - 1} + 1\) and \(g \circ f(x) = (\sqrt{x} + 1)^{2} - 1.\)

Step by step solution

01

1. Calculate \(f(g(x))\) #

To determine \(f(g(x))\), we will substitute the expression for \(g(x)\) into the function \(f(x)\): $$ f(g(x)) = f(x^{2} - 1). $$ Now, we will substitute \(x^{2} - 1\) for \(x\) in the function \(f(x) = \sqrt{x} + 1\): $$ f(g(x)) = \sqrt{x^{2} - 1} + 1. $$ So, the rule for the composite function \(f \circ g\) is: $$ f \circ g(x) = \sqrt{x^{2} - 1} + 1. $$
02

2. Calculate \(g(f(x))\) #

To determine \(g(f(x))\), we will substitute the expression for \(f(x)\) into the function \(g(x)\): $$ g(f(x)) = g(\sqrt{x} + 1). $$ Now, we will substitute \(\sqrt{x} + 1\) for \(x\) in the function \(g(x) = x^{2} - 1\): $$ g(f(x)) = (\sqrt{x} + 1)^{2} - 1. $$ So, the rule for the composite function \(g \circ f\) is: $$ g \circ f(x) = (\sqrt{x} + 1)^{2} - 1. $$ In conclusion, the composite functions are: $$ f \circ g(x) = \sqrt{x^{2} - 1} + 1, $$ and $$ g \circ f(x) = (\sqrt{x} + 1)^{2} - 1. $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Functions in Mathematics
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output. To visualize this, think of a function as a machine: you input a number, the machine processes it following a specific rule, and then outputs a new number. This rule is defined by the function expression. For example, with the function f(x) = ewlineewline + 1, for every x you input, the machine will output the square root of x and add 1 to it.

Functions can be represented in various forms including tables, graphs, or equations, and are fundamental in understanding relationships within many fields such as physics, engineering, and economics.
Function Composition
Function composition is the process of applying one function to the results of another. In essence, you're using the output from one function as the input for another function, creating a chain of functions. To express this, we use the notation f g(x), which means 'first apply g to x, then apply f to the result'. It is important to understand that the order in which functions are composed matters: f g(x) can yield a different result compared to g f(x). When composing functions, it's like giving the output of the first function to the second function as its meal; the second function then processes this meal according to its own recipe.
Square Root Functions
Square root functions are a type of radical function where the variable is under a square root. The basic form of a square root function is f(x) = {x}. When you plug in a number for x, the function outputs its square root. Imagine taking the square root as asking the question, 'What number, when multiplied by itself, gives the original number?' It's like finding the original side length of a square with an area equal to the number you're taking the square root of. Square root functions are used to solve equations and can also model real-world phenomena, such as calculating the side of a square given its area or understanding the intensity of an earthquake with the Richter scale.
Quadratic Functions
Quadratic functions are polynomial functions of degree two with the general form f(x) = ax^2 + bx + c, where a, b, and c are constants and a 0. The graph of a quadratic function is a parabola, which opens upwards if a > 0 and downwards if a < 0. These functions describe a multitude of natural and man-made phenomena, such as trajectories of objects under the influence of gravity or the area of a square when given the side length. Learning to work with quadratic functions is key to solving a wide range of problems in algebra and calculus.

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