- Isolate the term containing \(y\) in the equation \(25x^4 - 625y - 500 = 0\) by moving it to one side: \(625y = 25x^4 - 500\).
- Divide both sides by \(625\) to solve for \(y\): \(y = \frac{25x^4}{625} - \frac{500}{625}\).
- Simplify the fractions to obtain \(f(x) = \frac{1}{25}x^4 - \frac{4}{5}\).
- The correct option is \(\boxed{f(x) = \frac{1}{25}x^4 - \frac{4}{5}}\).
Explanation
- Isolating the term with \(y\)
We are given the equation \(25x^4 - 625y - 500 = 0\) and need to rewrite it as a function of \(x\), which means expressing \(y\) in terms of \(x\) as \(y = f(x)\). We start with the given equation:
\[25x^4 - 625y - 500 = 0\]
To isolate the term with \(y\), we add \(625y\) to both sides of the equation:
\[25x^4 - 500 = 625y\]
This is equivalent to:
\[625y = 25x^4 - 500\]
- Solving for \(y\)
Now, we need to solve for \(y\) by dividing both sides of the equation by \(625\):
\[y = \frac{25x^4 - 500}{625}\]
We can distribute the division by \(625\) to each term in the numerator:
\[y = \frac{25x^4}{625} - \frac{500}{625}\]
- Simplifying and selecting the correct answer
Next, we simplify each fraction. For the first term, we reduce \(\frac{25}{625}\):
\[\frac{25}{625} = \frac{25 \div 25}{625 \div 25} = \frac{1}{25}\]
For the second term, we reduce \(\frac{500}{625}\):
\[\frac{500}{625} = \frac{500 \div 125}{625 \div 125} = \frac{4}{5}\]
Substituting these simplified fractions back into the equation, we get:
\[y = \frac{1}{25}x^4 - \frac{4}{5}\]
Therefore, the function is \(f(x) = \frac{1}{25}x^4 - \frac{4}{5}\). Comparing this to the given options, it matches the first choice.
Examples
In physics, equations often relate several variables. For instance, when analyzing the motion of an object under a constant force, you might encounter an equation like \(F = ma\). If you know the force and mass, you rearrange it to find acceleration: \(a = F/m\). Similarly, this process of rearranging an equation to isolate one variable in terms of another is a fundamental skill in algebra used to graph relationships, solve for unknowns, or model systems in engineering and science.