Consider functions [tex]$f$[/tex] and [tex]$g$[/tex].
[tex]
\begin{array}{l}
f(x)=\sqrt{x^2+24 x+144} \\
g(x)=x^3-216
\end{array}
[/tex]
Which expression is equal to [tex]$f(x)+g(x)$[/tex]?
A. [tex]$x^3+x^2+24 x-72$[/tex]
B. [tex]$x^3+x-204$[/tex]
C. [tex]$x^4-204$[/tex]
D. [tex]$x^3+x-228$[/tex]
Answer (2)
Simplify f ( x ) to ∣ x + 12∣ .
Consider two cases: x ≥ − 12 and x < − 12 .
If x ≥ − 12 , then f ( x ) + g ( x ) = x 3 + x − 204 .
If x < − 12 , then f ( x ) + g ( x ) = x 3 − x − 228 .
Assuming x ≥ − 12 , the correct answer is x 3 + x − 204 .
Explanation
Simplify f(x) First, let's simplify the expression for f ( x ) . Notice that the expression inside the square root is a perfect square trinomial.
Rewrite f(x) We can rewrite f ( x ) as follows: f ( x ) = x 2 + 24 x + 144 = ( x + 12 ) 2 Since the square root of a square is the absolute value, we have: f ( x ) = ∣ x + 12∣
State g(x) Now, let's consider the function g ( x ) :
g ( x ) = x 3 − 216
Calculate f(x) + g(x) We want to find the expression for f ( x ) + g ( x ) . Since we have an absolute value in f ( x ) , we need to consider two cases:
Case 1: x ≥ − 12 . In this case, ∣ x + 12∣ = x + 12 , so f ( x ) = x + 12 .
Then, f ( x ) + g ( x ) = ( x + 12 ) + ( x 3 − 216 ) = x 3 + x + 12 − 216 = x 3 + x − 204 .
Case 2: x < − 12 . In this case, ∣ x + 12∣ = − ( x + 12 ) = − x − 12 , so f ( x ) = − x − 12 .
Then, f ( x ) + g ( x ) = ( − x − 12 ) + ( x 3 − 216 ) = x 3 − x − 12 − 216 = x 3 − x − 228 .
Compare with options Now, let's compare our results with the given options:
Option 1: x 3 + x 2 + 24 x − 72 Option 2: x 3 + x − 204 Option 3: x 4 − 204 Option 4: x 3 + x − 228
If x ≥ − 12 , then f ( x ) + g ( x ) = x 3 + x − 204 , which matches Option 2. If x < − 12 , then f ( x ) + g ( x ) = x 3 − x − 228 , which matches Option 4.
Select the correct answer Since the options do not specify a domain for x and Option 2 is present, we can assume that the intended domain is x ≥ − 12 . Therefore, the correct expression for f ( x ) + g ( x ) is x 3 + x − 204 .
Examples
Understanding function transformations and combinations is crucial in many fields. For example, in physics, you might combine functions to model the motion of an object under the influence of multiple forces. If one force is described by f ( x ) representing air resistance and another by g ( x ) representing gravitational force, the total force acting on the object would be f ( x ) + g ( x ) . By analyzing this combined function, you can predict the object's trajectory and behavior. Similarly, in economics, combining cost and revenue functions can help determine profit margins and optimize business strategies. Suppose the cost function is given by f ( x ) and the revenue function is given by g ( x ) , where x is the number of units sold. Then, the profit function is p ( x ) = g ( x ) − f ( x ) .
The function f ( x ) simplifies to ∣ x + 12∣ . Depending on the value of x , f ( x ) + g ( x ) can equal either x 3 + x − 204 for x ≥ − 12 or x 3 − x − 228 for x < − 12 . Assuming x ≥ − 12 , the correct option is B: x 3 + x − 204 .
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