At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

The function [tex][tex]$f$[/tex][/tex] is defined by [tex][tex]$f(x) = a \sqrt{x+b}$[/tex][/tex], where [tex][tex]$a$[/tex][/tex] and [tex][tex]$b$[/tex][/tex] are constants. In the [tex][tex]$xy$[/tex][/tex]-plane, the graph of [tex][tex]$y=f(x)$[/tex][/tex] passes through the point [tex][tex]$(-24,0)$[/tex][/tex], and [tex][tex]$f(24)\ \textless \ 0$[/tex][/tex]. Which of the following must be true?

A. [tex]f(0) = 24[/tex]
B. [tex]f(0) = -24[/tex]
C. [tex]a \ \textgreater \ b[/tex]
D. [tex]a \ \textless \ b[/tex]


Sagot :

Given the function [tex]\( f(x) = a \sqrt{x + b} \)[/tex] and the points through which the graph passes, we can determine the nature of the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex].

First, let's use the information that the graph passes through the point [tex]\((-24, 0)\)[/tex]. This means that when [tex]\( x = -24 \)[/tex], [tex]\( f(x) = 0 \)[/tex].

[tex]\[ f(-24) = a \sqrt{-24 + b} = 0 \][/tex]

Since [tex]\( \sqrt{-24 + b} = 0 \)[/tex], we have:

[tex]\[ -24 + b = 0 \implies b = 24 \][/tex]

So, we have determined that [tex]\( b = 24 \)[/tex].

Next, we were given that [tex]\( f(24) < 0 \)[/tex]. Let's use this condition to understand the nature of [tex]\( a \)[/tex].

[tex]\[ f(24) = a \sqrt{24 + b} = a \sqrt{24 + 24} = a \sqrt{48} \][/tex]

Since [tex]\( \sqrt{48} \)[/tex] is positive, for [tex]\( f(24) < 0 \)[/tex], it must be that [tex]\( a \)[/tex] is negative. Thus,

[tex]\[ a < 0 \][/tex]

With [tex]\( b = 24 \)[/tex] and [tex]\( a < 0 \)[/tex], let's now evaluate the options provided:

- Option (A) [tex]\( f(0) = 24 \)[/tex]:

[tex]\[ f(0) = a \sqrt{0 + b} = a \sqrt{24} \][/tex]

Since [tex]\( a < 0 \)[/tex], [tex]\( a \sqrt{24} \)[/tex] cannot be 24 because it would be a positive number if [tex]\( a \)[/tex] were positive, which is not the case here. So, this option is incorrect.

- Option (B) [tex]\( f(0) = -24 \)[/tex]:

[tex]\[ f(0) = a \sqrt{0 + b} = a \sqrt{24} \][/tex]

We need to check if this equals -24:

[tex]\[ a \sqrt{24} = -24 \implies a = -\frac{24}{\sqrt{24}} = -\sqrt{24} \][/tex]

Given [tex]\( \sqrt{24} \)[/tex] is positive and [tex]\( a = -\sqrt{24} \)[/tex] is indeed negative, so [tex]\( f(0) = -24 \)[/tex] is a valid option.

- Option (C) [tex]\( a > b \)[/tex]:

We know [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex]. Clearly [tex]\( a \)[/tex] is not greater than [tex]\( b \)[/tex] because [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex]. So this option is incorrect.

- Option (D) [tex]\( a < b \)[/tex]:

As [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex], it is true that [tex]\( a < b \)[/tex].

Thus, the options that are true based on the given conditions are:
- [tex]\( f(0) = -24 \)[/tex]
- [tex]\( a < b \)[/tex]

Thus the correct answers must include:

(B) [tex]\( f(0) = -24 \)[/tex]
(D) [tex]\( a < b \)[/tex]