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 truth of given statements.
The function is decreasing because the base 3 1 is between 0 and 1.
The y -intercept is found by setting x = 0 , resulting in y = ( 3 1 ) 0 = 1 , so the y -intercept is ( 0 , 1 ) .
The correct statements are that the function is decreasing and the y -intercept is ( 0 , 1 ) .
The final answer is: The function is decreasing and the y -intercept is ( 0 , 1 ) .
The function is decreasing and the y-intercept is (0,1)
Explanation
Analyzing the Problem We are given the function y = ( 3 1 ) x and asked to determine which statements about its graph are true. Let's analyze each statement.
Checking if the function is decreasing A function is decreasing if its value decreases as x increases. Since the base 3 1 is between 0 and 1, the function is decreasing. So, the statement 'The function is decreasing' is true.
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 never equals 0 for any real value of x . Therefore, there is no x -intercept. So, the statement 'The x -intercept is ( 1 , 0 ) ' is false.
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 : y = ( 3 1 ) 0 = 1 So, the y -intercept is ( 0 , 1 ) . Therefore, the statement 'The y -intercept is ( 0 , 1 ) ' is true.
Determining the range of the function The range of an exponential function of the form y = a x where 0 < a < 1 is all positive real numbers. In this case, the range is 0"> y > 0 . So, the statement 'The range of the function is all real numbers' is false.
Final Answer In summary, the true statements are:
The function is decreasing.
The y -intercept is ( 0 , 1 ) .
Examples
Exponential decay, similar to the function y = ( 3 1 ) x , is commonly used to model various real-world phenomena. For instance, it can describe the decrease in the amount of a radioactive substance over time, the cooling of an object, or the depreciation of an asset. Understanding exponential decay helps in making predictions and informed decisions in fields like physics, finance, and engineering. For example, if you invest in a stock that loses half its value each year, you can use an exponential decay model to estimate its future worth.
The true statements about the function y = ( 3 1 ) x are that it is decreasing and its y-intercept is (0,1). The function does not have an x-intercept and the range is all positive real numbers. Therefore, only the statements about the function being decreasing and the y-intercept being (0,1) are true.
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