Susan's science class is performing an experiment that involves dropping objects from various heights, starting close to the ground and working upward to 8 feet. The function [tex]$t(x)=\frac{1}{4} \sqrt{x}$[/tex], where [tex]$x$[/tex] represents the distance from the ground, represents the time it takes for the object Susan drops to hit the ground. The graph represents the function [tex]$t(x)=\frac{1}{4} \sqrt{x}$[/tex].
Identify some of the key features of the graph. That is, determine if the function is monotonically increasing or decreasing, state the end behavior, find the [tex]$x$[/tex]- and [tex]$y$[/tex]-intercepts, find the maximum or minimum, and state the domain and the range of the graph (without considering the context).
Answer (2)
The function t ( x ) = 4 1 x is monotonically increasing.
The x - and y -intercepts are both 0.
The domain and range are both [ 0 , ∞ ) .
The function has a minimum value of 0 and no maximum value (without context), and its end behavior approaches infinity as x approaches infinity. M o n o t o ni c a ll y I n cre a s in g , x -intercept: 0 , y -intercept: 0 , Domain: [ 0 , ∞ ) , Range: [ 0 , ∞ )
Explanation
Understanding the Problem We are given the function t ( x ) = 4 1 x , which represents the time it takes for an object to hit the ground when dropped from a height x . We need to analyze the key features of this function, including its monotonicity, end behavior, intercepts, maximum/minimum, domain, and range.
Monotonicity To determine if the function is monotonically increasing or decreasing, we can analyze its behavior. As x increases, x also increases, and thus 4 1 x increases. Therefore, the function is monotonically increasing. We can confirm this by evaluating the function at two points: t ( 1 ) = 0.25 and t ( 4 ) = 0.5 . Since t(1)"> t ( 4 ) > t ( 1 ) when 1"> 4 > 1 , the function is increasing.
x -intercept To find the x -intercept, we set t ( x ) = 0 and solve for x : 4 1 x = 0 x = 0 x = 0 So, the x -intercept is 0.
y -intercept To find the y -intercept, we set x = 0 and solve for t ( x ) : t ( 0 ) = 4 1 0 = 0 So, the y -intercept is 0.
End Behavior To determine the end behavior, we consider what happens to t ( x ) as x approaches infinity. As x → ∞ , x → ∞ , so t ( x ) = 4 1 x → ∞ .
Minimum Value To find the minimum value, we consider the domain of the function. Since x represents the distance from the ground, x ≥ 0 . The function is monotonically increasing, so the minimum value occurs at the smallest value of x , which is x = 0 . Thus, the minimum value is t ( 0 ) = 0 .
Maximum Value Since the function is monotonically increasing and we are not considering any upper bound for x (without context), there is no maximum value. The function increases without bound as x increases.
Domain The domain of the function t ( x ) = 4 1 x is all non-negative real numbers, since we can only take the square root of non-negative numbers. So, the domain is x ≥ 0 , or in interval notation, [ 0 , ∞ ) .
Range The range of the function t ( x ) = 4 1 x is all non-negative real numbers, since the square root function always returns non-negative values, and 4 1 times a non-negative value is also non-negative. So, the range is t ( x ) ≥ 0 , or in interval notation, [ 0 , ∞ ) .
Final Answer In summary, the function t ( x ) = 4 1 x is monotonically increasing, has an x -intercept of 0, a y -intercept of 0, its end behavior approaches infinity as x approaches infinity, has a minimum value of 0, has no maximum value (without context), a domain of [ 0 , ∞ ) , and a range of [ 0 , ∞ ) .
Examples
Understanding the properties of functions like t ( x ) = 4 1 x is crucial in various real-world scenarios. For instance, in physics, this function could model the time it takes for an object to fall a certain distance under constant acceleration. Knowing the domain and range helps determine the possible distances and corresponding times. Analyzing monotonicity helps understand how the time changes with increasing distance. This knowledge can be applied to design experiments, predict outcomes, and optimize processes in fields like engineering and sports science.
The function t ( x ) = 4 1 x is monotonically increasing with both x - and y -intercepts at 0. The domain and range are both [ 0 , ∞ ) , with a minimum value of 0 and no maximum value. As x approaches infinity, the function also approaches infinity.
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