Chapter 1.

1.1 Problem Statement

(6,8)
(2,4)
-$a*$b

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

(f º g)(x) = f(g(x))

, where a = hlF2UPykPxE=

Incorrect
Correct

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

(g º f)(x) = g(f(x))

, where b = iSba6t70dtA=

Incorrect
Correct

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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2
Correct.
Incorrect.

Question 1.4

For g(x), the denominator is x$b. This is zero when x = 1Wh3cvJ2xF4=.

Correct.
Incorrect.

Question 1.5

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Correct.
Incorrect.

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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Correct.
Incorrect.

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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Correct.
Incorrect.