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))
Sure! I can help you check your answers.
For the first question:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, the function is a linear function with no restrictions on the x-values. Therefore, your answer of the domain being (-∞, +∞) is correct.
For the second question:
To find the domain of the function g(x), we need to consider where the denominator (x^2 - 49) becomes zero because division by zero is undefined. The denominator becomes zero when x equals -7 and 7. Thus, the domain consists of all real numbers except -7 and 7. Your answer of (-∞, -7) U (-7, 7) U (7, +∞) is correct.
Now let's move on to finding the domain of the composite function f * g.
For the third question:
The composite function f * g means we need to substitute the function g(x) into f(x). So, we have f(g(x)) = f(1/(x+7)). To determine the domain of the composite function, we need to consider where the intermediate function g(x) is defined, as well as where the composed function f(g(x)) is defined.
First, let's look at the domain of g(x), which is (-∞, -7) U (-7, 7) U (7, +∞).
Then, we need to examine the composed function f(g(x)) = f(1/(x+7)). The function f(x) = x+3 is defined for all real numbers, so there are no restrictions on the range of g(x) that would prevent the composed function from being defined.
Therefore, the domain of the composite function f * g would be the same as the domain of g(x), which is (-∞, -7) U (-7, 7) U (7, +∞). Your answer of (-∞, -7) U (-7, 7) U (7, +∞) is correct.
For the fourth question:
Similarly, to find the domain of the composite function f * g, we need to substitute the function g(x) into f(x), which gives us f(g(x)) = f(x+2).
The function f(x) = 7x+70 is defined for all real numbers. In this case, there are no restrictions on the range of g(x) that would prevent the composed function from being defined.
Therefore, the domain of the composite function f * g would be the same as the domain of g(x), which is (-∞, -12) U (-12, +∞). Your answer of (-∞, -12) U (-12, +∞) is correct.
Great job! If you have any more questions, feel free to ask.