What is the composite function of
3x^2-5x+6 and x^2+3x and what is its domain and range
the answer depends on the composition. If
f(x) = 3x^2+5x+6
g(x) = x^2+3x
then
(f◦g)(x) = f(g(x))
= 3g^2-5g+6
= 3(x^2+3x)^2 - 5(x^2+3x) + 6
= 3x^4 + 18x^3 + 22x^2 - 15x + 6
(g◦f)(x) = g(f(x))
= f^2+3f
= (3x^2-5x+6)^2 + 3(3x^2-5x+6)
= 9x^4 - 30x^3 + 70x^2 - 75x + 54
In either case, the domain is all reals.
Since they are 4th-degree polynomials, they will have a minimum value. Finding that may not be easy, but the range will be (minval,+∞)
How about this idk if im right for this part
F(x)=2^x g(x)=y-x
Edit for g(x) i mean g(x)=3-x
To find the composite function of two functions, f(x) and g(x), we need to substitute g(x) into f(x). In this case, the functions are:
f(x) = 3x^2 - 5x + 6
g(x) = x^2 + 3x
To find the composite function, we substitute g(x) into f(x):
f(g(x)) = 3(g(x))^2 - 5(g(x)) + 6
Now, we substitute the expression for g(x):
f(g(x)) = 3(x^2 + 3x)^2 - 5(x^2 + 3x) + 6
Expanding and simplifying:
f(g(x)) = 3(x^4 + 6x^3 + 9x^2) - 5x^2 - 15x + 6
f(g(x)) = 3x^4 + 18x^3 + 27x^2 - 5x^2 - 15x + 6
f(g(x)) = 3x^4 + 18x^3 + 22x^2 - 15x + 6
Now let's discuss the domain and range of the composite function:
The domain of a composite function consists of all values of x that are within the domain of the inner function (g(x)) and make the composition well-defined. In this case, since both functions involve polynomials, there are no restrictions or excluded values for x. Therefore, the domain of the composite function is all real numbers (-∞, ∞).
The range of a composite function can be found by analyzing the possible output values of the function. In this case, since both f(x) and g(x) are polynomials, their ranges are all real numbers. Thus, the range of the composite function is also all real numbers (-∞, ∞).