Questions: The functions f and g are defined as f(x)=x^3 and g(x)=4x^2+17x-15. 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).

The functions f and g are defined as f(x)=x^3 and g(x)=4x^2+17x-15. 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 $f(x)=x^{3}$ and $g(x)=4 x^{2}+17 x-15$. a) Find the domain of $f, g, f+g, f-g, f g, f f, \frac{f}{g}$, and $\frac{g}{f}$. b) Find $(f+g)(x)$, $(f-g)(x)$, $(f g)(x)$, $(f f)(x)$, $\left(\frac{f}{g}\right)(x)$, and $\left(\frac{g}{f}\right)(x)$.
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Solution

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Solution Steps

Step 1: Perform the operation

The operation f+g on the functions x^3 and 4_x^2 + 17_x - 15 gives: $x^3 + 4_x^2 + 17_x - 15$

Step 2: Determine the domain

The domain of the resulting function is all real numbers unless the operation introduces additional restrictions.

Final Answer:

The result of the operation f+g is $x^3 + 4_x^2 + 17_x - 15$ with domain all real numbers unless the operation introduces additional restrictions., rounded to 2 decimal places.

Step 1: Perform the operation

The operation f-g on the functions x^3 and 4_x^2 + 17_x - 15 gives: $x^3 - 4_x^2 - 17_x + 15$

Step 2: Determine the domain

The domain of the resulting function is all real numbers unless the operation introduces additional restrictions.

Final Answer:

The result of the operation f-g is $x^3 - 4_x^2 - 17_x + 15$ with domain all real numbers unless the operation introduces additional restrictions., rounded to 2 decimal places.

Step 1: Perform the operation

The operation fg on the functions x^3 and 4_x^2 + 17_x - 15 gives: $x^3_(4_x^2 + 17*x - 15)$

Step 2: Determine the domain

The domain of the resulting function is all real numbers unless the operation introduces additional restrictions.

Final Answer:

The result of the operation fg is $x^3_(4_x^2 + 17*x - 15)$ with domain all real numbers unless the operation introduces additional restrictions., rounded to 2 decimal places.

Step 1: Perform the operation

The operation ff on the functions x^3 and 4_x^2 + 17_x - 15 gives: $x^9$

Step 2: Determine the domain

The domain of the resulting function is all real numbers unless the operation introduces additional restrictions.

Final Answer:

The result of the operation ff is $x^9$ with domain all real numbers unless the operation introduces additional restrictions., rounded to 2 decimal places.

Step 1: Perform the operation

The operation f/g on the functions x^3 and 4_x^2 + 17_x - 15 gives: $x^3/(4_x^2 + 17_x - 15)$

Step 2: Determine the domain

The domain of the resulting function is excluding values where the denominator is zero: [-5, 3/4]

Final Answer:

The result of the operation f/g is $x^3/(4_x^2 + 17_x - 15)$ with domain excluding values where the denominator is zero: [-5, 3/4], rounded to 2 decimal places.

Step 1: Perform the operation

The operation g/f on the functions x^3 and 4_x^2 + 17_x - 15 gives: $(4_x^2 + 17_x - 15)/x^3$

Step 2: Determine the domain

The domain of the resulting function is excluding values where the denominator is zero: [0]

Final Answer:

The result of the operation g/f is $(4_x^2 + 17_x - 15)/x^3$ with domain excluding values where the denominator is zero: [0], rounded to 2 decimal places.

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