What is the y-intercept of $f(x)=\left(\frac{1}{4}\right)^x$?
A. $(0,1)$
B. $\left(1, \frac{1}{4}\right)$
C. $(0,0)$
D. $(1,0)$
Answer (1)
To find the y-intercept, set x = 0 in the function f ( x ) = ( 4 1 ) x .
Calculate f ( 0 ) = ( 4 1 ) 0 = 1 .
The y-intercept is the point ( 0 , f ( 0 )) .
The y-intercept is ( 0 , 1 ) .
Explanation
Understanding the Problem We are asked to find the y-intercept of the function f ( x ) = ( 4 1 ) x . The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when x = 0 .
Finding the y-intercept To find the y-intercept, we need to evaluate the function at x = 0 . That is, we need to calculate f ( 0 ) .
Calculating f(0) We have f ( 0 ) = ( 4 1 ) 0 . Any non-zero number raised to the power of 0 is 1. Therefore, f ( 0 ) = 1 .
Stating the Answer The y-intercept is the point ( 0 , f ( 0 )) , which is ( 0 , 1 ) . Therefore, the correct answer is A. ( 0 , 1 ) .
Examples
Imagine you're tracking the decay of a radioactive substance. The function f ( x ) = ( 4 1 ) x could represent the fraction of the substance remaining after x units of time. The y-intercept, ( 0 , 1 ) , tells you that at the beginning (time x = 0 ), you have 100% of the substance. Understanding y-intercepts helps you determine initial conditions in various real-world scenarios, such as population growth, compound interest, or the depreciation of an asset.
