Hi,
I had a question about exponential functions. If f(x)=ab to the x power, were a does not equal zero, b is greater than 0 and b does not equal one.
Therefore if b=1/2, then the graph of the function never crosses the y-axis. What is the answer to this question, true or false and why? Please help me.
If f(x) = (ab)^x and b = 1/2,
f(x) = (a/2)^x
The y-axis is where x = 0. y = f(x) would equal (a/2)^0 there, which means y=1.
The statement is false. For any real, postive a, (a/2)^0 = 1
To determine whether the statement is true or false, we need to understand the behavior of the exponential function f(x)=ab^x, where a ≠ 0, b > 0, and b ≠ 1.
Let's focus on the condition b = 1/2. If b = 1/2, it means that the base of the exponential function is less than 1. When the base of an exponential function is less than 1, the function starts at a positive value and decreases rapidly as x increases.
Now, let's think about what it means for the graph to cross the y-axis. Crossing the y-axis occurs when the value of the function is zero, which means that f(x) = 0.
For our given function f(x) = ab^x where b = 1/2, to find the x-value where f(x) equals zero, we need to solve the equation:
0 = ab^x
Since a ≠ 0 and b ≠ 0, we can divide both sides of the equation by a to get:
0/a = b^x
Since b ≠ 0, we can conclude that any number raised to the power of x will never equal zero.
Hence, the statement that the graph of the function never crosses the y-axis when b = 1/2 is TRUE. This is because when the base of the exponential function is less than 1, it will not reach zero; it will approach zero but will never actually reach it.
So, in summary, the graph of the function f(x) = ab^x, where b = 1/2, never intersects or crosses the y-axis.