Select The Correct Answer.Rewrite The Following Equation As A Function Of $x$.$5x^4 - 625y - 500 = 0$A. \$f(x) = 25x^4 - \frac{5}{4}$[/tex\] B. $f(x) = \frac{1}{25}x^4 - \frac{4}{5}$ C. $f(x) =

by ADMIN 204 views

Introduction

In mathematics, equations are a fundamental concept that helps us understand the relationship between variables. When we are given an equation, we often need to rewrite it in a specific form to make it easier to solve or analyze. In this article, we will focus on rewriting the given equation as a function of $x$.

The Given Equation

The given equation is:

5x4−625y−500=05x^4 - 625y - 500 = 0

Our goal is to rewrite this equation as a function of $x$.

Step 1: Isolate the Variable

To rewrite the equation as a function of $x$, we need to isolate the variable $x$. We can start by moving the constant term to the right-hand side of the equation.

5x4−625y=5005x^4 - 625y = 500

Step 2: Add 625y to Both Sides

Next, we add $625y$ to both sides of the equation to get rid of the negative term.

5x4=500+625y5x^4 = 500 + 625y

Step 3: Divide Both Sides by 5

Now, we divide both sides of the equation by 5 to isolate the term with $x$.

x4=500+625y5x^4 = \frac{500 + 625y}{5}

Step 4: Simplify the Right-Hand Side

We can simplify the right-hand side of the equation by dividing the numerator and denominator by 5.

x4=100+125yx^4 = 100 + 125y

Step 5: Rewrite the Equation as a Function of x

Now that we have isolated the variable $x$, we can rewrite the equation as a function of $x$.

f(x)=100+125yf(x) = 100 + 125y

However, we need to express $y$ in terms of $x$. Since the original equation is $5x^4 - 625y - 500 = 0$, we can solve for $y$.

−625y=−5x4+500-625y = -5x^4 + 500

y=−5x4+500−625y = \frac{-5x^4 + 500}{-625}

y=5x4−500625y = \frac{5x^4 - 500}{625}

Now, we can substitute this expression for $y$ into the equation for $f(x)$.

f(x)=100+125(5x4−500625)f(x) = 100 + 125\left(\frac{5x^4 - 500}{625}\right)

Simplifying the Expression

We can simplify the expression for $f(x)$ by multiplying the terms inside the parentheses.

f(x)=100+125(5x4−500)625f(x) = 100 + \frac{125(5x^4 - 500)}{625}

f(x)=100+625x4−62500625f(x) = 100 + \frac{625x^4 - 62500}{625}

f(x)=100+625x4625−62500625f(x) = 100 + \frac{625x^4}{625} - \frac{62500}{625}

f(x)=100+x4−100f(x) = 100 + x^4 - 100

f(x)=x4f(x) = x^4

Conclusion

In conclusion, the correct answer is:

f(x)=x4f(x) = x^4

This is the rewritten equation as a function of $x$. We started with the given equation $5x^4 - 625y - 500 = 0$ and isolated the variable $x$ by moving the constant term to the right-hand side and dividing both sides by 5. We then simplified the right-hand side and expressed $y$ in terms of $x$. Finally, we substituted this expression for $y$ into the equation for $f(x)$ and simplified the resulting expression.

Answer Options

Let's compare our answer with the given options:

A. $f(x) = 25x^4 - \frac{5}{4}$

B. $f(x) = \frac{1}{25}x^4 - \frac{4}{5}$

C. $f(x) = x^4$

Q: What is the main goal of rewriting the equation as a function of x?

A: The main goal of rewriting the equation as a function of x is to isolate the variable x and express it in terms of a function.

Q: How do I isolate the variable x in the given equation?

A: To isolate the variable x, you need to move the constant term to the right-hand side of the equation and then divide both sides by the coefficient of x.

Q: What is the difference between the original equation and the rewritten equation?

A: The original equation is $5x^4 - 625y - 500 = 0$, while the rewritten equation is $f(x) = x^4$. The rewritten equation is a function of x, while the original equation is not.

Q: Why do we need to express y in terms of x?

A: We need to express y in terms of x because the original equation is a function of both x and y. By expressing y in terms of x, we can rewrite the equation as a function of x alone.

Q: How do I simplify the expression for f(x)?

A: To simplify the expression for f(x), you need to multiply the terms inside the parentheses and then combine like terms.

Q: What is the final answer to the problem?

A: The final answer to the problem is $f(x) = x^4$.

Q: Why is option C the correct answer?

A: Option C is the correct answer because it matches the rewritten equation $f(x) = x^4$.

Q: What are some common mistakes to avoid when rewriting an equation as a function of x?

A: Some common mistakes to avoid when rewriting an equation as a function of x include:

  • Not isolating the variable x
  • Not expressing y in terms of x
  • Not simplifying the expression for f(x)
  • Not checking the final answer against the original equation

Q: How do I check my answer against the original equation?

A: To check your answer against the original equation, you need to substitute the expression for f(x) back into the original equation and verify that it is true.

Q: What are some real-world applications of rewriting an equation as a function of x?

A: Some real-world applications of rewriting an equation as a function of x include:

  • Modeling population growth
  • Analyzing financial data
  • Solving optimization problems
  • Predicting future trends

Q: How do I apply the concept of rewriting an equation as a function of x to real-world problems?

A: To apply the concept of rewriting an equation as a function of x to real-world problems, you need to:

  • Identify the variables and constants in the problem
  • Isolate the variable x
  • Express y in terms of x
  • Simplify the expression for f(x)
  • Verify the final answer against the original equation

By following these steps, you can apply the concept of rewriting an equation as a function of x to a wide range of real-world problems.