How do you find the domain and range of #y = 4x - x ^2#?

Answer 1

Domain: #(-oo, +oo)#
Range: #(-oo, 4]#

#y= 4x-x^2#
#y# is defined #forall x in RR# Hence the domain of #y# is #(-oo, +oo)#
Since the coefficient of #x^2# is -ve, y has a maximum value where #y'=0#
#y' = 4 -2x=0#
#x=2 -> y_max =4*2 - 2^2 =4#
Thus the range of #y# is: #(-oo, 4]#
This can be observed from the graph of #y# below.

plot{x(4-x) [-6.4, 6.087, -1.347, 4.897]}

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Answer 2

To find the domain and range of the function y = 4x - x^2, we need to analyze the behavior of the function.

  1. Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since the function is a polynomial, there are no restrictions on the input values. Therefore, the domain is all real numbers, or (-∞, ∞).

  2. Range: The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range, we can analyze the behavior of the function. Since the function is a quadratic function with a negative coefficient for the x^2 term, it opens downward. Therefore, the maximum value of the function occurs at the vertex. The x-coordinate of the vertex of the quadratic function y = ax^2 + bx + c is given by x = -b/(2a). In this case, a = -1 and b = 4. Plugging these values into the formula, we get x = -4/(2*(-1)) = 2. To find the corresponding y-coordinate (maximum value), we plug x = 2 into the function: y = 4(2) - 2^2 = 8 - 4 = 4. Therefore, the maximum value of the function is 4. Since the function is a quadratic function, it approaches negative infinity as x approaches positive or negative infinity. Thus, the range of the function is (-∞, 4].

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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