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Chapter 1: Problem 13

Graph each function with a graphing utility using the given window. Then state the domain and range of the function. $$f(x)=3 x^{4}-10 ; \quad[-2,2] \times[-10,15]$$

Short Answer

Expert verified
Question: Graph the function $$f(x) = 3x^4 - 10$$ with the given window $$[-2, 2] \times [-10, 15]$$ and state its domain and range. Answer: The domain of the function $$f(x) = 3x^4 - 10$$ is $(-\infty, \infty)$. The range of the function is $(-10, \infty)$.

Step by step solution

01

Graph the function using a graphing utility

Use a graphing utility (like Desmos, Geogebra, or a graphing calculator) to graph the function $$f(x)=3x^4 -10$$ with the given window $$[-2,2]\times[-10,15]$$. The graph helps us visualize the function, and the window restricts the graph to the specified interval.
02

Determine the domain of the function

Domain represents the range of allowed input values (x-values) for the function. Since $$f(x) = 3x^4 - 10$$ is a polynomial function, it is defined for all real numbers. Therefore, the domain would be $$(-\infty, \infty)$$.
03

Determine the range of the function using the graph

The range represents the range of resulting output values (y-values) for the function. Looking at the graph of the function, we can see that it has a minimum value (the lowest point on the graph) and that the function continuously increases as x goes to the right or left. The minimum value occurs at x=0, which is $$f(0)=3(0)^4 -10 = -10$$. Since the function increases without bound as x goes to the left or right, the range of the function is $$(-10, \infty)$$.

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