Questions: Question 5 (5 points) For each of the following combined functions, choose the option which describes how to find its domain. f + g fg f - g f / g g o f 1. Find the domain of f and the domain of g separately, regardless of how the new combined function looks. 2. Find the domain of f and the domain of g separately, and throw out any x-values which cause g to output 0, regardless of how the new combined function looks. 3. Find the domain of f, and throw out any x-values which cause f to output values that are not permitted by g, regardless of how the new combined function looks. Question 6 (1 point) Consider the function shown below; assume that there is not graph to the left of the y-axis, and that consecutive tick marks along the y-axis shows 1 unit.

Question 5 (5 points)
For each of the following combined functions, choose the option which describes how to find its domain.
f + g
fg
f - g
f / g
g o f
1. Find the domain of f and the domain of g separately, regardless of how the new combined function looks.
2. Find the domain of f and the domain of g separately, and throw out any x-values which cause g to output 0, regardless of how the new combined function looks.
3. Find the domain of f, and throw out any x-values which cause f to output values that are not permitted by g, regardless of how the new combined function looks.

Question 6 (1 point)
Consider the function shown below; assume that there is not graph to the left of the y-axis, and that consecutive tick marks along the y-axis shows 1 unit.

Transcript text: Question 5 (5 points) For each of the following combined functions, choose the option which describes how to find its domain. $\mathrm{f}+\mathrm{g}$ $\mathrm{f}\mathrm{g}$ $\mathrm{f}-\mathrm{g}$ $\mathrm{f} / \mathrm{g}$ $\mathrm{g}\mathrm{o}\mathrm{f}$ 1. Find the domain of $f$ and the domain of g separately, regardless of how the new combined function looks. 2. Find the domain of $f$ and the domain of g separately, and throw out any $x$-values which cause $g$ to output 0, regardless of how the new combined function looks. 3. Find the domain of $f$, and throw out any $x$-values which cause $f$ to output values that are not permitted by g, regardless of how the new combined function looks. Question 6 (1 point) Consider the function shown below; assume that there is not graph to the left of the $y$-axis, and that consecutive tick marks along the $y$-axis shows 1 unit.

Solution

Solution Steps

Solution Approach

For Question 5:

For the combined function $ f + g $, the domain is the intersection of the domains of $ f $ and $ g $.
For the combined function $ f \cdot g $, the domain is also the intersection of the domains of $ f $ and $ g $.
For the combined function $ f - g $, the domain is the intersection of the domains of $ f $ and $ g $.
For the combined function $ \frac{f}{g} $, the domain is the intersection of the domains of $ f $ and $ g $, excluding any $ x $-values that make $ g(x) = 0 $.
For the combined function $ g \circ f $ (composition of $ g $ and $ f $), the domain is the set of all $ x $-values in the domain of $ f $ such that $ f(x) $ is in the domain of $ g $.

For Question 6: To determine the domain of the function based on the given information, we need to analyze the graph and identify the range of $ x $-values for which the function is defined.

Step 1: Determine the Domain of $ f + g $

The domain of $ f + g $ is the intersection of the domains of $ f $ and $ g $. Given: \[ \text{domain}_f = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] \[ \text{domain}_g = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] The intersection is: \[ \text{domain}_{f+g} = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

Step 2: Determine the Domain of $ f \cdot g $

The domain of $ f \cdot g $ is also the intersection of the domains of $ f $ and $ g $: \[ \text{domain}_{f \cdot g} = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

Step 3: Determine the Domain of $ f - g $

The domain of $ f - g $ is the intersection of the domains of $ f $ and $ g $: \[ \text{domain}_{f - g} = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

Step 4: Determine the Domain of $ \frac{f}{g} $

The domain of $ \frac{f}{g} $ is the intersection of the domains of $ f $ and $ g $, excluding any $ x $-values that make $ g(x) = 0 $: \[ \text{domain}_{\frac{f}{g}} = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{0\} \] \[ \text{domain}_{\frac{f}{g}} = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

Step 5: Determine the Domain of $ g \circ f $

The domain of $ g \circ f $ is the set of all $ x $-values in the domain of $ f $ such that $ f(x) $ is in the domain of $ g $: \[ \text{domain}_{g \circ f} = \{x \in \text{domain}_f \mid f(x) \in \text{domain}_g\} \] Given the example functions, the domain is: \[ \text{domain}_{g \circ f} = \{-3, -2, -1, 0, 1, 2, 3\} \]