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Chapter 6: Problem 41

Verify that the functions \(f\) and g are inverses of each other by showing that \(f(g(x))=x\) and \(g(f(x))=x\). Give any values of x that need to be excluded from the domain of \(f\) and the domain of g. $$ f(x)=\frac{2 x+3}{x+4} ; \quad g(x)=\frac{4 x-3}{2-x} $$

Short Answer

Expert verified
The functions are inverses. Exclude \(x = -4\) from the domain of \(f(x)\) and \(x = 2\) from the domain of \(g(x)\).

Step by step solution

01

Compute \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\). Start by finding \(f\left(g(x)\right)\). Given: \(f(x)=\frac{2x+3}{x+4}\) and \(g(x)=\frac{4x-3}{2-x}\). Substitute \(g(x)\) for \(x\) in \(f(x)\):\[f(g(x)) = f\left(\frac{4x-3}{2-x}\right) = \frac{2\left(\frac{4x-3}{2-x}\right) + 3}{\left(\frac{4x-3}{2-x}\right) + 4}\]Simplify the numerator and the denominator.
02

Simplify the Numerator of f(g(x))

Simplify the numerator: \[2\left(\frac{4x-3}{2-x}\right) + 3 = \frac{8x-6}{2-x} + \frac{6x-12}{2-x} = \frac{8x-6 + 6(2-x)}{2-x} = \frac{8x-6 + 12 - 6x}{2-x} = \frac{2x + 6}{2-x}\]
03

Simplify the Denominator of f(g(x))

Simplify the denominator: \[\left(\frac{4x-3}{2-x}\right) + 4 = \frac{4x-3}{2-x} + \frac{8-4x}{2-x} = \frac{4x-3 + 8-4x}{2-x} = \frac{5}{2-x}\]
04

Complete f(g(x))

Complete the function: \[f(g(x)) = \frac{\frac{2x+6}{2-x}}{\frac{5}{2-x}} = \frac{2x+6}{5} = x\]
05

Compute g(f(x))

Now substitute \(f(x)\) into \(g(x)\). Start by finding \(g\left(f(x)\right)\). Given: \(f(x)=\frac{2x+3}{x+4}\) and \(g(x)=\frac{4x-3}{2-x}\). Substitute \(f(x)\) for \(x\) in \(g(x)\):\[g(f(x)) = g\left(\frac{2x+3}{x+4}\right) = \frac{4\left(\frac{2x+3}{x+4}\right)-3}{2-\left(\frac{2x+3}{x+4}\right)}\]Simplify the numerator and the denominator.
06

Simplify the Numerator of g(f(x))

Simplify the numerator:\[4\left(\frac{2x+3}{x+4}\right)-3 = \frac{8x+12}{x+4}-3 = \frac{8x+12 - 3(x+4)}{x+4} = \frac{8x+12-3x-12}{x+4} = \frac{5x}{x+4}\]
07

Simplify the Denominator of g(f(x))

Simplify the denominator:\[2-\left(\frac{2x+3}{x+4}\right) = \frac{2(x+4)-2x-3}{x+4} = \frac{2x+8-2x-3}{x+4} = \frac{5}{x+4}\]
08

Complete g(f(x))

Complete the function:\[g(f(x)) = \frac{\frac{5x}{x+4}}{\frac{5}{x+4}} = x\]
09

Identify Exclusions from the Domain

Identify any values that must be excluded from the domain. For \(f(x)\), the denominator must not be zero since division by zero is undefined. Thus, exclude \(x = -4\). For \(g(x)\), the denominator must also not be zero, thus exclude \(x = 2\).

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Key Concepts

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

Inverse Functions
Inverse functions are pairs of functions where one function reverses the effect of the other. In simpler terms, if you have a function, say f, and another function, say g, then g is the inverse of f if applying f followed by g returns you to your original value. If you have f then g and it gets you back to the same starting number, then you have an inverse relationship between f and g. To mathematically confirm this relationship, we compute the composition of these functions, i.e., we calculate f(g(x)) and g(f(x)), and check if both are equal to x.
Domain Exclusions
When working with functions, there are certain values of x that you must avoid, especially those that make the denominator zero because division by zero is undefined. For example, in the given functions, the denominator of f(x) = (2x+3)/(x+4) cannot be zero, so we must exclude x = -4 from the domain of f(x). Similarly, for g(x) = (4x-3)/(2-x), we must exclude x = 2 from the domain of g(x). To remember it easily, always identify the values that make denominators zero and exclude them from your domain.
Function Composition
Function composition involves applying one function to the result of another function. If you have two functions f and g, the composition f(g(x)) means you first apply g to x, then apply f to the result of g(x). To verify that two functions are inverses, we need to show that f(g(x)) equals x and g(f(x)) also equals x. This process requires substituting one function into the other and simplifying the expressions rigorously until you either confirm or refute their inverseness.
Rational Functions
Rational functions are fractions involving polynomials. For example, f(x) = (2x+3)/(x+4) and g(x) = (4x-3)/(2-x) are both rational functions. These types of functions often have restrictions on their domains due to their denominators not being zero. A key fact about rational functions is their behavior at these points and how they approach infinity. When simplifying composition of rational functions, it's essential to combine and factor polynomials carefully and correctly handle the common denominator.

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