Chapter 1.
1.1 Problem Statement
Calculate the composite functions f º g and g º f, and determine their domains.
1.2 Step 1
To calculate (f º g)(x) = f(g(x)), substitue into the expression for f(x) and simplify the expression to a single fraction.
Question Sequence
Question
1.1
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1.3 Step 2
Similarly, calculate (g º f)(x) = g(f(x)) by substituting into the expression for g(x) and simplify the expression.
Question Sequence
Question
1.2
, where b = iSba6t70dtA=
1.4 Step 3
To obtain the domain of either f º g or g º f, we need to find the domain of f(x) and g(x) individually.
Since f(x) and g(x) are both rational functions, the only x-values that need to be excluded are those that make the denominator zero.
For f(x), the denominator is x$a + 1 and is never zero.
Question Sequence
Question
1.3
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Question
1.4
For g(x), the denominator is x$b. This is zero when x = 1Wh3cvJ2xF4=.
Question
1.5
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1.5 Step 4
Since (f º g)(x) = f(g(x)), f(x) takes as inputs the values from g(x) to obtain the domain for (f º g)(x),we need to intersect the domain of g(x) with the domain of the expression for (f º g)(x).
Now (f º g)(x) = is a rational function so that the only x-values that need to be excluded are those that make the denominator zero or undefined. The denominator of (f º g)(x) is x$ab+1, which is never zero.
Question
1.6
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1.6 Step 5
Since (g º f)(x) = g(f(x)), g(x) takes, as inputs, the values from f(x), to find the domain for (g º f)(x) we need to intersect the domain of f(x) with the domain of the expression for (g º f)(x).
Now (g º f)(x) = (x$a + 1)$b is a polynomial function, which is defined for all real numbers.
Question Sequence
Question
1.7
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Question
1.8
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