Let f(x) = x^2 + 5 and g(x) = x – 7. Perform each function operation. Then find the domain of each.
1. f(x)/g(x)
2. f(x)-2g(x)
3. f(x)*g(x)
1. f(x)/g(x) = (x^2 + 5)/(x - 7)
The domain of f(x)/g(x) is all real numbers except x=7, because division by zero is undefined. So the domain is: x ∈ ℝ, x ≠ 7.
2. f(x) - 2g(x) = (x^2 + 5) - 2(x - 7) = x^2 + 5 - 2x + 14 = x^2 - 2x + 19
The domain of f(x)-2g(x) is all real numbers, because there are no restrictions on the values of x for this expression. So the domain is: x ∈ ℝ.
3. f(x)*g(x) = (x^2 + 5)(x - 7) = x^3 - 7x^2 + 5x - 35
The domain of f(x)*g(x) is all real numbers because there are no restrictions on the values of x for this expression. So the domain is: x ∈ ℝ.
1. To perform the operation f(x)/g(x), we need to divide the function f(x) by the function g(x).
f(x)/g(x) = (x^2 + 5)/(x - 7)
To find the domain of this function, we need to consider the values of x for which the denominator (x - 7) is not equal to zero, since division by zero is undefined.
Setting the denominator equal to zero:
x - 7 = 0
x = 7
So, the only value that makes the denominator zero is x = 7. Therefore, the domain of f(x)/g(x) is all real numbers except x = 7.
Domain: (-∞, 7) U (7, +∞)
2. To perform the operation f(x) - 2g(x), we need to subtract twice the function g(x) from the function f(x).
f(x) - 2g(x) = x^2 + 5 - 2(x - 7)
= x^2 + 5 - 2x + 14
= x^2 - 2x + 19
Since there are no restrictions or forbidden values for this expression, the domain is all real numbers.
Domain: (-∞, +∞)
3. To perform the operation f(x) * g(x), we need to multiply the function f(x) by the function g(x).
f(x) * g(x) = (x^2 + 5) * (x - 7)
= x^3 - 7x^2 + 5x - 35
Again, there are no restrictions or forbidden values for this expression, so the domain is all real numbers.
Domain: (-∞, +∞)
To perform each function operation, we will substitute the given function expressions into the respective operations and simplify them.
1. f(x)/g(x):
Substituting the values of f(x) and g(x) into the division operation, we get:
f(x)/g(x) = (x^2 + 5)/(x - 7)
To find the domain of f(x)/g(x), we need to look for any values of x that would make the denominator (x - 7) equal to zero. In this case, x = 7 would make the denominator zero, which is not allowed. Hence, the domain of f(x)/g(x) would be all real numbers except x = 7.
2. f(x) - 2g(x):
Substituting the values of f(x) and g(x) into the subtraction operation, we get:
f(x) - 2g(x) = (x^2 + 5) - 2(x - 7)
Simplifying further:
f(x) - 2g(x) = x^2 + 5 - 2x + 14
= x^2 - 2x + 19
There are no denominators involved in this expression, so there are no restrictions on the domain. Hence, the domain of f(x) - 2g(x) is all real numbers.
3. f(x) * g(x):
Substituting the values of f(x) and g(x) into the multiplication operation, we get:
f(x) * g(x) = (x^2 + 5)(x - 7)
To find the domain of f(x) * g(x), we need to consider any values of x that would make either of the factors equal to zero. In this case, there are no such values that would make either factor zero. Hence, there are no restrictions on the domain of f(x) * g(x), and it is defined for all real numbers.