The function $f(x)=a^x-4$ will never cross the $x$-axis if $a$ is positive.
A. True
B. False
Answer (1)
Set f ( x ) = 0 to find where the function crosses the x-axis: a x − 4 = 0 .
Solve for x : a x = 4 , which gives x = l n ( a ) l n ( 4 ) for a = 1 .
Since a is positive (and not equal to 1), there exists a real number x such that f ( x ) = 0 .
Therefore, the statement is false: False .
Explanation
Understanding the Problem We are given the function f ( x ) = a x − 4 and asked to determine if it will never cross the x-axis when a is positive. Crossing the x-axis means there exists a value of x such that f ( x ) = 0 .
Setting up the Equation To find where the function crosses the x-axis, we set f ( x ) = 0 and solve for x :
a x − 4 = 0 a x = 4
Solving for x To solve for x , we take the logarithm of both sides. We can use any base for the logarithm, but we'll use the natural logarithm (base e ) for convenience: ln ( a x ) = ln ( 4 ) x ln ( a ) = ln ( 4 ) If a e q 1 , we can divide by ln ( a ) to get: x = ln ( a ) ln ( 4 )
Considering the case a=1 If a = 1 , then the equation a x = 4 becomes 1 x = 4 , which simplifies to 1 = 4 . This is a contradiction, so a cannot be equal to 1.
Analyzing the Solution Since a is positive and a e q 1 , the expression x = l n ( a ) l n ( 4 ) is a real number. This means that for any positive a (except a = 1 ), there exists a real number x such that f ( x ) = 0 . Therefore, the function f ( x ) will cross the x-axis.
Final Answer The statement
Conclusion The function f ( x ) = a x − 4 will never cross the x -axis if a is positive
Final Answer is false, because we found that for any positive a (except 1), there is an x such that f ( x ) = 0 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population starts at 4 million and grows exponentially, the function f ( x ) = a x − 4 can represent the population's deviation from a certain threshold (4 million) over time. Understanding when this function crosses the x-axis (i.e., when f ( x ) = 0 ) helps us determine when the population reaches that threshold. This concept is crucial in environmental science, finance, and other fields where exponential models are applied.
