Question
How many roots do the functions have in common?
\[f(x)=x^2-4x-5\]
A coordinate plane. The x- and y-axes both scale by one. The graph is the function y equals g of x which is a parabola that opens up. The function has a y-intercept at zero, nine, an x-intercept at two, zero, a minimum around three point five, negative two, and another x-intercept at five, zero.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[y=g(x)\]
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[f\] and
\[g\] share the same root(s).
A
\[f\] and
\[g\] share the same root(s).
(Choice B)
\[f\] and
\[g\] share one root in common but each have another root that is not shared.
B
\[f\] and
\[g\] share one root in common but each have another root that is not shared.
(Choice C)
\[f\] and
\[g\] share no roots in common.
C
\[f\] and
\[g\] share no roots in common.
\[f(x)=x^2-4x-5\]
A coordinate plane. The x- and y-axes both scale by one. The graph is the function y equals g of x which is a parabola that opens up. The function has a y-intercept at zero, nine, an x-intercept at two, zero, a minimum around three point five, negative two, and another x-intercept at five, zero.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[y=g(x)\]
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[f\] and
\[g\] share the same root(s).
A
\[f\] and
\[g\] share the same root(s).
(Choice B)
\[f\] and
\[g\] share one root in common but each have another root that is not shared.
B
\[f\] and
\[g\] share one root in common but each have another root that is not shared.
(Choice C)
\[f\] and
\[g\] share no roots in common.
C
\[f\] and
\[g\] share no roots in common.
Answers
GPT-4o mini
To find how many roots the functions \( f(x) \) and \( g(x) \) have in common, let's first determine the roots of function \( f(x) \).
The function \( f(x) = x^2 - 4x - 5 \) can be factored or solved using the quadratic formula. We can factor it as follows:
\[
f(x) = (x - 5)(x + 1)
\]
This gives us the roots:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, \( f(x) \) has roots \( x = 5 \) and \( x = -1 \).
Next, we examine the function \( g(x) \) based on the description given. \( g(x) \) is a parabola that opens up, has a y-intercept at \( (0, 9) \), and has two x-intercepts at approximately \( (2, 0) \) and \( (5, 0) \). Therefore, it appears that the roots of \( g(x) \) are \( x = 2 \) and \( x = 5 \).
Now, let's summarize the roots we found:
- The roots of \( f(x) \): \( -1, 5 \)
- The roots of \( g(x) \): \( 2, 5 \)
The common root between the two functions is \( x = 5 \).
Thus, \( f \) and \( g \) share **one root** in common but each have another root that is not shared.
The correct answer is:
**Choice B**: \( f \) and \( g \) share one root in common but each have another root that is not shared.
The function \( f(x) = x^2 - 4x - 5 \) can be factored or solved using the quadratic formula. We can factor it as follows:
\[
f(x) = (x - 5)(x + 1)
\]
This gives us the roots:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, \( f(x) \) has roots \( x = 5 \) and \( x = -1 \).
Next, we examine the function \( g(x) \) based on the description given. \( g(x) \) is a parabola that opens up, has a y-intercept at \( (0, 9) \), and has two x-intercepts at approximately \( (2, 0) \) and \( (5, 0) \). Therefore, it appears that the roots of \( g(x) \) are \( x = 2 \) and \( x = 5 \).
Now, let's summarize the roots we found:
- The roots of \( f(x) \): \( -1, 5 \)
- The roots of \( g(x) \): \( 2, 5 \)
The common root between the two functions is \( x = 5 \).
Thus, \( f \) and \( g \) share **one root** in common but each have another root that is not shared.
The correct answer is:
**Choice B**: \( f \) and \( g \) share one root in common but each have another root that is not shared.