Questions: The functions f and g are defined as f(x)=3x+4 and g(x)=3-10x. a) Find the domain of f, g, f+g, f-g, fg, ff, f/g, and g/f. b) Find (f+g)(x), (f-g)(x), (fg)(x), (ff)(x), (f/g)(x), and (g/f)(x).

Transcript text: The functions $f$ and $g$ are defined as $f(x)=3 x+4$ and $g(x)=3-10 x$. a) Find the domain of $f, g, f+g, f-g, f g, f f, \frac{f}{g}$, and $\frac{g}{f}$. b) Find $(f+g)(x),(f-g)(x),(f g)(x)$, (ff)(x), $\left(\frac{\mathrm{f}}{\mathrm{g}}\right)(\mathrm{x})$, and $\left(\frac{\mathrm{g}}{\mathrm{f}}\right)(\mathrm{x})$.

Solution

Solution Steps

Step 1: Define the operation

The operation given is 'add'. The functions are $f(x) = 3x + 4$ and $g(x) = -10x + 3$.

Step 2: Perform the operation

The result of the operation is: \[(3-10)x + (4+3)\] Simplified, this is: \[-7x + 7\]

Step 3: Determine the domain

The domain of the resultant function is: All real numbers, $\mathbb{R}$.

Final Answer:

The simplified result of the operation is \[-7x + 7\] with domain All real numbers, $\mathbb{R}$.

Step 1: Define the operation

The operation given is 'subtract'. The functions are $f(x) = 3x + 4$ and $g(x) = -10x + 3$.

Step 2: Perform the operation

The result of the operation is: \[(3+10)x + (4-3)\] Simplified, this is: \[13x + 1\]

Step 3: Determine the domain

The domain of the resultant function is: All real numbers, $\mathbb{R}$.

Final Answer:

The simplified result of the operation is \[13x + 1\] with domain All real numbers, $\mathbb{R}$.

Step 1: Define the operation

The operation given is 'multiply'. The functions are $f(x) = 3x + 4$ and $g(x) = -10x + 3$.

Step 2: Perform the operation

The result of the operation is: \[(3_-10)x^2 + (3_3+4_-10)x + 4_3\] Simplified, this is: \[-30x^2 - 31x + 12\]

Step 3: Determine the domain

The domain of the resultant function is: All real numbers, $\mathbb{R}$.

Final Answer:

The simplified result of the operation is \[-30x^2 - 31x + 12\] with domain All real numbers, $\mathbb{R}$.

Step 1: Define the operation

The operation given is 'multiply'. The functions are $f(x) = 3x + 4$ and $g(x) = 3x + 4$.

Step 2: Perform the operation

The result of the operation is: \[(3_3)x^2 + (3_4+4_3)x + 4_4\] Simplified, this is: \[9x^2 + 24x + 16\]

Step 3: Determine the domain

The domain of the resultant function is: All real numbers, $\mathbb{R}$.

Final Answer:

The simplified result of the operation is \[9x^2 + 24x + 16\] with domain All real numbers, $\mathbb{R}$.

Step 1: Define the operation

The operation given is 'divide_f_by_g'. The functions are $f(x) = 3x + 4$ and $g(x) = -10x + 3$.

Step 2: Perform the operation

The result of the operation is: \[\frac{3x+4}{-10x+3}\]

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