h(x)=x^3 -4. Find the domain, range and intercepts
all polynomials have domain of all real numbers.
All polynomials of odd degree have range of all real numbers,
y=0 when x = ∛4
x=0 when y = -4
Find the domain, range, and intercepts for h(x) = x^3 -4
To find the domain, range, and intercepts of the function h(x) = x^3 - 4, let's go step by step:
1. Domain: The domain of a function represents all the possible x-values for which the function is defined. Since a cubic function like h(x) = x^3 - 4 is defined for all real numbers, the domain is (-∞, ∞), meaning it includes all real numbers.
2. Range: The range of a function represents all the possible y-values that the function can take. For a cubic function, the range is also all real numbers, as the function can approach positive infinity or negative infinity depending on the behavior of the cubic term x^3. Therefore, the range is (-∞, ∞).
3. x-intercepts: To find the x-intercepts, set h(x) equal to zero and solve for x. In this case, we have:
x^3 - 4 = 0
x^3 = 4
Taking the cube root of both sides, we get:
x = ∛4
Hence, the x-intercept is x = ∛4 (which is approximately 1.5874).
4. y-intercept: The y-intercept represents the value of y when x is zero. Substituting x = 0 into the function, we have:
h(0) = (0)^3 - 4
h(0) = -4
Therefore, the y-intercept is y = -4.
In summary, the domain of the function is (-∞, ∞), the range is (-∞, ∞), the x-intercept is approximately 1.5874, and the y-intercept is -4.