State the domain and range for following:
A) y=4x
B) y=12
C) y=x^2-4x+1
D) y=x^2+2x+1
E) y=3x^3-x^2-1
F) y=3x^3+x^2-1
To find the domain and range of each function, we need to consider the values that x can take for the domain and the corresponding values of y for the range.
A) For y=4x, the domain can be any real number since there are no restrictions on x. So, the domain for this function is (-∞, ∞) or all real numbers. The range is also (-∞, ∞) because when we substitute any real number for x, we get a corresponding real number for y.
B) In the function y=12, x does not affect the value of y since it is a constant. As a result, the domain is still (-∞, ∞) or all real numbers. However, the output or range of this function is a single value, which is 12. So, the range is {12}.
C) For y=x^2-4x+1, the domain can be any real number since there are no restrictions on x. So, the domain for this function is (-∞, ∞) or all real numbers. To find the range, we need to analyze the graph or plot some points from the function. However, we can also use the fact that the function represents a parabola. Since the coefficient of x^2 is positive, the parabola opens upward and has a minimum value. Therefore, the range for this function is [-(D/4), ∞), where D is the discriminant or the value under the square root in the quadratic formula. In this case, D=16-4(1)(1) = 12. Thus, the range is [-3, ∞).
D) For y=x^2+2x+1, the domain can still be any real number since there are no restrictions on x. So, the domain for this function is (-∞, ∞). Again, this is a quadratic function, and since the coefficient of x^2 is positive, the parabola opens upward. Thus, the range is [-(D/4), ∞). In this case, D=2^2 - 4(1)(1) = 0. Therefore, the range is [0, ∞).
E) In y=3x^3-x^2-1, the domain is still (-∞, ∞) since there are no restrictions on x. To find the range, we can analyze the behavior of the function. As x approaches positive infinity or negative infinity, y also tends to positive or negative infinity, respectively. Therefore, the range is (-∞, ∞).
F) Finally, for y=3x^3+x^2-1, the domain is still (-∞, ∞) due to no restrictions on x. The range, however, can be calculated similarly to the previous function. As x approaches positive infinity or negative infinity, y also tends to positive or negative infinity, respectively. Hence, the range is (-∞, ∞).