Could you please help me and check my answers?

Find the domain of the function.

f(x)=-8x-2
-I got the domain being (negative infinity, postive infinity).

g(x)= 3x/x^2 -49 (I got (-inifinity,-7)U (-7,7) U (7, +ininfinity) The U is the union symbol.

Find the domain of the composite function f * g.

12. f(x)=x+3, g(x)=1/(x+7) I got (-inifinity, + inifinity) but would it be ((-inifinity,-7) or (-7,+inifinity)

15. f(x)=7x+70, g(x)=x+2 I got for this one ((-infinity,-12) or (-12,+inifinity))

To find the domain of a function, you need to identify the values of x for which the function is defined.

For the function f(x) = -8x - 2, since this is a linear function, there are no restrictions on the domain. Therefore, the domain is (-∞, +∞), which means all real numbers.

For the function g(x) = 3x / (x^2 - 49), there are two conditions to consider. First, the denominator should not be zero, as division by zero is undefined. Second, the value under the square root should not be negative.

To find the domain, we need to consider the values that make the denominator zero and the values that cause the expression under the square root to be negative.

For the denominator, we have (x^2 - 49) ≠ 0. Factoring the denominator, we get (x - 7)(x + 7) ≠ 0. This means x cannot be equal to -7 or 7.

Next, for the expression under the square root, we have (x^2 - 49) ≥ 0. Factorizing further, we get (x - 7)(x + 7) ≥ 0. This inequality is true for values of x ≤ -7 and x ≥ 7.

Combining these two conditions, we find the domain of g(x) to be (-∞, -7] ∪ [-7, 7) ∪ (7, +∞).

Moving on to the composite function f * g, which is denoted as (f ∘ g)(x), we need to evaluate f(g(x)). In other words, we first substitute g(x) into f(x).

Given f(x) = x + 3 and g(x) = 1/(x + 7), we substitute g(x) into f(x): f(g(x)) = g(x) + 3 = (1/(x + 7)) + 3.

To find the domain of the composite function, we need to consider the values of x that make the expression (1/(x + 7)) + 3 defined. Since we have a fraction, the denominator (x + 7) should not be zero. Therefore, x cannot be equal to -7.

Thus, the domain of the composite function f * g is (-∞, -7) ∪ (-7, +∞).

For the function f(x) = 7x + 70 and g(x) = x + 2, there are no restrictions on the domain of either function. Therefore, the domain of f(x) and g(x) is (-∞, +∞).

Thus, the domain of the composite function f * g is also (-∞, +∞).