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Use intercepts to graph each equation. 6x-9y-18 = 0
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 69
Find a. (fog) (x) b. the domain of f o g. f(x) = x/(x+1), g(x) = 4/x
Guida verificata passo dopo passo1
First, recall that the composition of functions (fog)(x) means f(g(x)). So, we need to substitute g(x) into the function f.
Given f(x) = \(\frac{x}{x+1}\) and g(x) = \(\frac{4}{x}\), substitute g(x) into f(x) to get (fog)(x) = f\(\left\)(\(\frac{4}{x}\)\(\right\)) = \(\frac{\frac{4}{x}\)}{\(\frac{4}{x}\) + 1}.
Simplify the expression for (fog)(x) by combining the terms in the denominator: \(\frac{4}{x}\) + 1 = \(\frac{4}{x}\) + \(\frac{x}{x}\) = \(\frac{4 + x}{x}\). Then rewrite (fog)(x) as \(\frac{\frac{4}{x}\)}{\(\frac{4 + x}{x}\)}.
To simplify the complex fraction, multiply numerator and denominator by x to eliminate the denominators inside the fraction: (fog)(x) = \(\frac{4}{4 + x}\).
For the domain of (fog), consider the domain restrictions from both f and g. First, g(x) = \(\frac{4}{x}\) requires x \(\neq\) 0. Next, f(g(x)) requires the denominator of f(g(x)) to be nonzero, so 4 + x \(\neq\) 0, which means x \(\neq\) -4. Therefore, the domain of (fog) is all real numbers except x = 0 and x = -4.
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Function Composition
Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). It requires substituting the entire expression of g(x) into f(x), which helps in combining two functions into a single expression.
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Percorso guidato
4:56
Function Composition
Domain of a Function
The domain of a function is the set of all input values (x) for which the function is defined. When composing functions, the domain of (f ∘ g) includes all x-values in the domain of g such that g(x) is in the domain of f.
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3:51Domain Restrictions of Composed Functions
Restrictions from Denominators
Functions with variables in denominators have restrictions where the denominator cannot be zero. Identifying these values is crucial to determine the domain, especially when composing functions, to avoid division by zero.
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02:58Rationalizing Denominators
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