Which function has the greatest $x$-intercept?

$f(x)=3 x-9$
$g(x)=|x+3|$
$h(x)=2^x-16$
$j(x)=-5(x-2)^2$

Find the x-intercept of f ( x ) = 3 x − 9 by solving 3 x − 9 = 0 , which gives x = 3 .
Find the x-intercept of g ( x ) = ∣ x + 3∣ by solving ∣ x + 3∣ = 0 , which gives x = − 3 .
Find the x-intercept of h ( x ) = 2 x − 16 by solving 2 x − 16 = 0 , which gives x = 4 .
Find the x-intercept of j ( x ) = − 5 ( x − 2 ) 2 by solving − 5 ( x − 2 ) 2 = 0 , which gives x = 2 . The greatest x-intercept is 4 ​ .

Explanation

Understanding the Problem 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 the value of the function is zero at that point. So, we need to find the value of x for which each function equals zero.

Finding x-intercept of f(x) Let's find the x-intercept of f ( x ) = 3 x − 9 . We set f ( x ) = 0 and solve for x : 3 x − 9 = 0 3 x = 9 x = 3 9 ​ x = 3

Finding x-intercept of g(x) Now, let's find the x-intercept of g ( x ) = ∣ x + 3∣ . We set g ( x ) = 0 and solve for x : ∣ x + 3∣ = 0 x + 3 = 0 x = − 3

Finding x-intercept of h(x) Next, let's find the x-intercept of h ( x ) = 2 x − 16 . We set h ( x ) = 0 and solve for x : 2 x − 16 = 0 2 x = 16 Since 16 = 2 4 , we have: 2 x = 2 4 x = 4

Finding x-intercept of j(x) Finally, let's find the x-intercept of 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

Comparing the x-intercepts We have found the x-intercepts of the four functions: f ( x ) has an x-intercept of 3. g ( x ) has an x-intercept of -3. h ( x ) has an x-intercept of 4. j ( x ) has an x-intercept of 2. The greatest x-intercept is 4, which belongs to the function h ( x ) .


Examples
Understanding x-intercepts is crucial in many real-world applications. For instance, in business, the x-intercept of a cost function can represent the break-even point, where the company's expenses equal its revenue. Similarly, in physics, the x-intercept of a projectile's trajectory can indicate the distance the projectile travels before hitting the ground. Knowing how to find and interpret x-intercepts allows us to analyze and make informed decisions in various practical scenarios.

Answered by GinnyAnswer | 2025-07-05

The greatest x-intercept among the functions is from h ( x ) = 2 x − 16 , which is 4. The other x-intercepts are 3 for f ( x ) , -3 for g ( x ) , and 2 for j ( x ) . Therefore, the answer is 4.
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Answered by Anonymous | 2025-07-07