Which statements about the graph of the function $y=\left(\frac{1}{3}\right)^x$ are true?
* The function is increasing.
* The function is decreasing.
* The $x$-intercept is $(1,0)$.
* The $y$-intercept is $(0,1)$.
* The range of the function is all real numbers.
Answer (2)
The function y = ( 3 1 ) x is analyzed to determine the truthfulness of given statements.
It is determined that the function is decreasing because as x increases, y decreases.
The y -intercept is found by setting x = 0 , resulting in y = 1 , so the y -intercept is ( 0 , 1 ) .
The true statements are that the function is decreasing and the y -intercept is ( 0 , 1 ) . The final answer is: Decreasing, y -intercept is ( 0 , 1 ) .
Decre a s in g , y − in t erce pt i s ( 0 , 1 )
Explanation
Analyzing the Problem We are given the function y = ( 3 1 ) x and asked to determine which of the given statements about its graph are true. Let's analyze each statement.
Checking if the function is decreasing A function is decreasing if, as x increases, y decreases. Since the base 3 1 is between 0 and 1, the function is decreasing. To confirm, let's pick two values for x , say x 1 = 1 and x 2 = 2 . Then y 1 = ( 3 1 ) 1 = 3 1 and y 2 = ( 3 1 ) 2 = 9 1 . Since \frac{1}{9}"> 3 1 > 9 1 , the function is decreasing.
Finding the x-intercept The x -intercept is the point where the graph intersects the x -axis, which means y = 0 . We need to solve the equation ( 3 1 ) x = 0 . However, an exponential function of the form y = a x (where 0"> a > 0 and a = 1 ) never equals zero. Therefore, there is no x -intercept.
Finding the y-intercept The y -intercept is the point where the graph intersects the y -axis, which means x = 0 . We need to find the value of y when x = 0 . So, y = ( 3 1 ) 0 = 1 . Thus, the y -intercept is ( 0 , 1 ) .
Determining the range of the function The range of an exponential function of the form y = a x (where 0"> a > 0 and a = 1 ) is all positive real numbers. In this case, the range is 0"> y > 0 . Therefore, the range is not all real numbers.
Conclusion Based on our analysis:
The function is decreasing.
The x -intercept is not ( 1 , 0 ) .
The y -intercept is ( 0 , 1 ) .
The range of the function is not all real numbers.
Therefore, the true statements are:
The function is decreasing.
The y -intercept is ( 0 , 1 ) .
Examples
Exponential decay, similar to the function in the problem, is used to model various real-world phenomena such as the decay of radioactive substances, the cooling of an object, or the depreciation of an asset. For example, if a car's value depreciates by 3 2 each year, its value can be modeled by the function V ( t ) = V 0 ( 3 1 ) t , where V 0 is the initial value and t is the time in years. Understanding the properties of exponential functions helps in predicting the future value of the car or the remaining amount of a radioactive substance.
The function y = ( 3 1 ) x is decreasing and has a y-intercept at ( 0 , 1 ) . It does not have an x-intercept, and its range is all positive real numbers, not all real numbers. The true statements are that the function is decreasing and the y-intercept is ( 0 , 1 ) .
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