Select the correct answer.
Rewrite the following equation as a function of $x$.
[tex]25 x^4-625 y-500=0[/tex]
A. [tex]f(x)=\frac{1}{25} x^4+\frac{4}{5}[/tex]
B. [tex]f(x)=25 x^4-\frac{5}{4}[/tex]
C. [tex]f(x)=-25 x^{h 4}+\frac{5}{4}[/tex]
D. [tex]f(x)=\frac{1}{25} x^4-\frac{4}{5}[/tex]
Answer (2)
Isolate the term with y : 625 y = 25 x 4 − 500 .
Divide by 625 to solve for y : y = 625 25 x 4 − 500 .
Simplify the expression: y = 25 x 4 − 5 4 .
Express as a function: f ( x ) = 25 1 x 4 − 5 4 .
The correct answer is D .
Explanation
Understanding the problem We are given the equation 25 x 4 − 625 y − 500 = 0 and we want to rewrite it as a function of x , which means we want to express y in terms of x .
Isolating the y term First, let's isolate the term with y on one side of the equation. We can add 625 y to both sides to get:
25 x 4 − 500 = 625 y
Solving for y Now, we can divide both sides of the equation by 625 to solve for y :
y = 625 25 x 4 − 500
Simplifying the expression Next, we simplify the expression for y by dividing each term in the numerator by 625:
y = 625 25 x 4 − 625 500
Reducing the fractions Now, we reduce the fractions:
y = 25 x 4 − 5 4
Expressing as a function of x Finally, we express the result as a function of x :
f ( x ) = 25 1 x 4 − 5 4
Selecting the correct option Comparing this result with the given options, we see that it matches option D.
Examples
In physics, this type of equation can describe the trajectory of a particle. If you know how the particle's horizontal position changes over time (described by x ), you can find its vertical position ( y ) using this function. This is useful in fields like projectile motion, where understanding the path of an object is crucial.
To rewrite the equation 25 x 4 − 625 y − 500 = 0 as a function of x , we isolate y and simplify to obtain f ( x ) = 25 1 x 4 − 5 4 . This corresponds to option D in the multiple-choice answers. The steps involved include isolating the variable, simplifying, and writing in function notation.
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