The function f is defined by f(x) = a√x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) passes through the point (-24, 0), and f(24)

Understand the Problem

The question is asking which statement must be true about the function defined, given its properties and specific points it passes through. It involves evaluating relationships between the function's values at specific points and its constants.

Answer

$f(0) = -24$.

Steps to Solve

Identify the function and its properties
The function is defined as $f(x) = a\sqrt{x} + b$. We know that it passes through the point $(-24, 0)$, which tells us that $f(-24) = 0$. Thus, we will set up the equation from this point.
Set up the equation from the given point
Substituting $x = -24$ into the function:
$$ 0 = a\sqrt{-24} + b $$
However, since $\sqrt{-24}$ is not a real number, the point must be considered in the context that $\sqrt{x}$ is only defined for $x \geq 0$. This condition indicates the function cannot pass through the point $(-24, 0)$ and should be examined under real values for $x$.
Evaluate the condition $f(24) < 0$
Substituting $x = 24$ into the function:
$$ f(24) = a\sqrt{24} + b $$
For this to be less than zero:
$$ a\sqrt{24} + b < 0 $$
Thus, we have:
$$ b < -a\sqrt{24} $$
Determine the valid conditions for $b$ using $f(0)$
Substituting $x = 0$ gives:
$$ f(0) = a\sqrt{0} + b = b $$
Thus, $f(0) = b$. Now, we can analyze options with the relationship derived.

From the previous steps, $b < -a\sqrt{24}$ suggests we need to decide how $a$ and $b$ relate to satisfy this inequality while keeping in mind that $b$ must be defined from the limiting behaviour of $a$.

Analyze each statement
Evaluate each option in light of the conditions derived about $a$ and $b$:

A: $f(0) = 24 \implies b = 24$, which may contradict the conditions if $a > 0$.
B: $f(0) = -24 \implies b = -24$, which provides a more consistent possibility if $a$ is positive.
C: $a > b$, which does not necessarily hold with our conditions since we do not have direct information on either constant already.
D: $a < b$, which could hold depending on their relationship.

Evaluating these options leads us to conclude the necessary conditions for $a$ and $b$.

The statement that must be true is $B: f(0) = -24$.

More Information

The conditions come from the need for $a$ and $b$ to be correctly constrained such that $f(24) < 0$ holds while considering the effect of both parameters on the function.

Tips

Assuming $f(x)$ can take on values for $x < 0$ without considering the nature of the square root function.
Misinterpreting the implications of the inequality $f(24) < 0$ in terms of $a$ and $b$.