Which function has the greatest [tex]$x$[/tex]-intercept?
[tex]$f(x)=3 x-9$[/tex]
[tex]$g(x)=|x+3|$[/tex]
[tex]$h(x)=2^x-16$[/tex]
[tex]$j(x)=-5(x-2)^2$[/tex]
Answer (1)
Find the x-intercept of f ( x ) = 3 x − 9 : x = 3 .
Find the x-intercept of g ( x ) = ∣ x + 3∣ : x = − 3 .
Find the x-intercept of h ( x ) = 2 x − 16 : x = 4 .
Find the x-intercept of j ( x ) = − 5 ( x − 2 ) 2 : x = 2 .
The greatest x-intercept is 4 .
Explanation
Problem Analysis We are given four functions and we need to find the greatest x-intercept among them. The x-intercept is the point where the function crosses the x-axis, which means f ( x ) = 0 . We will find the x-intercept for each function and then compare them.
Finding x-intercept of f(x) For f ( x ) = 3 x − 9 , we set f ( x ) = 0 and solve for x :
3 x − 9 = 0 3 x = 9 x = 3 So, the x-intercept of f ( x ) is 3.
Finding x-intercept of g(x) For g ( x ) = ∣ x + 3∣ , we set g ( x ) = 0 and solve for x :
∣ x + 3∣ = 0 x + 3 = 0 x = − 3 So, the x-intercept of g ( x ) is -3.
Finding x-intercept of h(x) For h ( x ) = 2 x − 16 , we set h ( x ) = 0 and solve for x :
2 x − 16 = 0 2 x = 16 2 x = 2 4 x = 4 So, the x-intercept of h ( x ) is 4.
Finding x-intercept of j(x) For j ( x ) = − 5 ( x − 2 ) 2 , we set j ( x ) = 0 and solve for x :
− 5 ( x − 2 ) 2 = 0 ( x − 2 ) 2 = 0 x − 2 = 0 x = 2 So, the x-intercept of j ( x ) is 2.
Comparing x-intercepts and concluding Comparing the x-intercepts, we have 3, -3, 4, and 2. The greatest of these is 4, which corresponds to the function h ( x ) .
Examples
Understanding x-intercepts is crucial in many real-world applications. For example, in business, the x-intercept of a cost function can represent the break-even point, where the company starts making a profit. Similarly, in physics, the x-intercept of a projectile's trajectory can represent the distance the projectile travels before hitting the ground. By finding and comparing x-intercepts, we can make informed decisions and predictions in various fields.
