Which of these functions has exactly two different zeros? A function has exactly two different zeros when exactly two distinct real input values produce an output of zero. To answer that question, set each function equal to zero, solve for x, and count only unique real solutions. Multiplicity does not create an additional zero, and excluded domain values cannot qualify Which is the point..
What Does “Exactly Two Different Zeros” Mean?
A zero of a function is an input value that makes the function equal to zero. For a function written as y = f(x), its zeros satisfy:
f(x) = 0
Here's one way to look at it: if f(x) = x² - 4, then:
x² - 4 = 0
(x - 2)(x + 2) = 0
So:
x = 2 or x = -2
The function has two different zeros: -2 and 2. On a graph, these correspond to two distinct
Let’s look at each candidate one by one And it works..
First function: (f_{1}(x)= (x-2)^{2}).
Setting the expression equal to zero gives ((x-2)^{2}=0), which yields the single solution (x=2). Even though the root occurs twice, it counts as one distinct zero, so (f_{1}) has only one zero.
Second function: (f_{2}(x)= (x-1)(x-3)).
Solving ((x-1)(x-3)=0) produces the two real solutions (x=1) and (x=3). Both are different and lie within the domain (the function is defined for every real number), so (f_{2}) possesses exactly two distinct zeros That's the part that actually makes a difference. Less friction, more output..
Third function: (f_{3}(x)=\dfrac{1}{x-4}+2) The details matter here..
To find its zeros we set the expression equal to zero:
[ \frac{1}{x-4}+2=0 ;\Longrightarrow; \frac{1}{x-4}=-2 ;\Longrightarrow; x-4=-\frac12 ;\Longrightarrow; x=3.5. ]
The value (x=4) would make the denominator zero, but it is not a solution of the equation; the only admissible solution is (x=3.5). Hence (f_{3}) has just one zero Nothing fancy..
Conclusion
Among the three functions examined, only (f_{2}(x)= (x-1)(x-3)) meets the criterion of having exactly two different zeros. All other candidates either have a single distinct zero or a repeated root that does not create an additional distinct solution.
