Questions: The domain of f(x) is the set of all real values except 7, and the domain of g(x) is the set of all real values except -3. Which of the following describes the domain of (g ∘ f)(x)? - all real values except x=-3 and the x for which f(x) ≠ 7 - all real values except x ≠-3 and the x for which f(x)=-3 - all real values except x=7 and the x for which f(x)=7 - all real values except x ≠ 7 and the x for which f(x) ≠-3

The domain of f(x) is the set of all real values except 7, and the domain of g(x) is the set of all real values except -3. Which of the following describes the domain of (g ∘ f)(x)?
- all real values except x=-3 and the x for which f(x) ≠ 7
- all real values except x ≠-3 and the x for which f(x)=-3
- all real values except x=7 and the x for which f(x)=7
- all real values except x ≠ 7 and the x for which f(x) ≠-3

Transcript text: The domain of $f(x)$ is the set of all real values except 7, and the domain of $g(x)$ is the set of all real values except -3 . Which of the following describes the domain of $(g \circ f)(x)$ ? all real values except $x=-3$ and the $x$ for which $f(x) \neq 7$ all real values except $x \neq-3$ and the $x$ for which $f(x)=-3$ all real values except $x=7$ and the $x$ for which $f(x)=7$ all real values except $x \neq 7$ and the $x$ for which $f(x) \neq-3$

Solution

Solution Steps

To determine the domain of the composition function $(g \circ f)(x)$, we need to consider the domains of both $f(x)$ and $g(x)$. Specifically, $f(x)$ must be defined, and $g(f(x))$ must also be defined. This means $x$ must be in the domain of $f(x)$, and $f(x)$ must be in the domain of $g(x)$.

$f(x)$ is defined for all $x$ except $x = 7$.
$g(x)$ is defined for all $x$ except $x = -3$.

Therefore, for $(g \circ f)(x)$ to be defined:

$x$ must not be 7 (so $f(x)$ is defined).
$f(x)$ must not be -3 (so $g(f(x))$ is defined).

Thus, the domain of $(g \circ f)(x)$ is all real values except $x = 7$ and the $x$ for which $f(x) = -3$.

Step 1: Determine the Domain of $ f(x) $

The function $ f(x) $ is defined for all real values except $ x = 7 $. Therefore, the domain of $ f(x) $ is: \[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq 7 \} \]

Step 2: Determine the Domain of $ g(x) $

The function $ g(x) $ is defined for all real values except $ x = -3 $. Therefore, the domain of $ g(x) $ is: \[ \text{Domain of } g(x) = \{ x \in \mathbb{R} \mid x \neq -3 \} \]

Step 3: Determine the Domain of $ (g \circ f)(x) $

For the composition $ (g \circ f)(x) $ to be defined:

$ f(x) $ must be defined, which means $ x \neq 7 $.
$ g(f(x)) $ must be defined, which means $ f(x) \neq -3 $.

Thus, the domain of $ (g \circ f)(x) $ is all real values except $ x = 7 $ and the $ x $ for which $ f(x) = -3 $.

Step 4: Identify the Values of $ x $ for which $ f(x) = -3 $

Given $ f(x) = x^2 - 10 $, we solve for $ x $ when $ f(x) = -3 $: \[ x^2 - 10 = -3 \] \[ x^2 = 7 \] \[ x = \pm \sqrt{7} \]

Therefore, $ f(x) = -3 $ when $ x = \sqrt{7} $ or $ x = -\sqrt{7} $.

Final Answer

The domain of $ (g \circ f)(x) $ is all real values except $ x = 7 $ and $ x = \pm \sqrt{7} $. This corresponds to the second option in the multiple-choice question.

\[ \boxed{\text{all real values except } x = 7 \text{ and the } x \text{ for which } f(x) = -3} \]

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