To solve the problem, we will first define the functions you mentioned:
- \( f(x) = x^2 - 6x \)
- \( g(x) = 7 + x \)
Now, we can find the following:
(a) \( (f + g)(x) \)
The sum of the two functions is given by:
\[ (f + g)(x) = f(x) + g(x) \]
Calculating that:
\[ f(x) + g(x) = (x^2 - 6x) + (7 + x) \]
Combining like terms:
\[ = x^2 - 6x + x + 7 = x^2 - 5x + 7 \]
Thus:
\[ (f + g)(x) = x^2 - 5x + 7 \]
(b) \( (f - g)(x) \)
The difference of the two functions is given by:
\[ (f - g)(x) = f(x) - g(x) \]
Calculating that:
\[ f(x) - g(x) = (x^2 - 6x) - (7 + x) \]
Distributing the negative sign:
\[ = x^2 - 6x - 7 - x = x^2 - 7x - 7 \]
Thus:
\[ (f - g)(x) = x^2 - 7x - 7 \]
(c) \( (f \cdot g)(x) \)
The product of the two functions is given by:
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
Calculating that:
\[ f(x) \cdot g(x) = (x^2 - 6x)(7 + x) \]
Using the distributive property (FOIL):
\[ = x^2 \cdot 7 + x^2 \cdot x - 6x \cdot 7 - 6x \cdot x \]
Calculating each term:
\[ = 7x^2 + x^3 - 42x - 6x^2 \]
Combining like terms:
\[ = x^3 + (7x^2 - 6x^2) - 42x = x^3 + x^2 - 42x \]
Thus:
\[ (f \cdot g)(x) = x^3 + x^2 - 42x \]
(d) \( \left( \frac{f}{g} \right)(x) \)
The quotient of the two functions is given by:
\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
Calculating it:
\[ \left( \frac{f}{g} \right)(x) = \frac{x^2 - 6x}{7 + x} \]
Thus:
\[ \left( \frac{f}{g} \right)(x) = \frac{x^2 - 6x}{7 + x} \]
(e) The domain of \( \left( \frac{f}{g} \right) \)
The domain of a function is determined by the values of \( x \) for which the function is defined. For \( \left( \frac{f}{g} \right)(x) \), it is undefined when the denominator \( g(x) \) is equal to zero.
Setting the denominator to zero:
\[ 7 + x = 0 \]
Solving for \( x \):
\[ x = -7 \]
Thus, the domain of \( \left( \frac{f}{g} \right) \) is all real numbers except \( -7 \), which can be written as:
\[ \text{Domain} = { x \in \mathbb{R} \mid x \neq -7 } \] or in interval notation:
\[ (-\infty, -7) \cup (-7, \infty) \]
In summary:
- (a) \( (f + g)(x) = x^2 - 5x + 7 \)
- (b) \( (f - g)(x) = x^2 - 7x - 7 \)
- (c) \( (f \cdot g)(x) = x^3 + x^2 - 42x \)
- (d) \( \left( \frac{f}{g} \right)(x) = \frac{x^2 - 6x}{7 + x} \)
- (e) Domain: \( (-\infty, -7) \cup (-7, \infty) \)
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