Which equation is the inverse of 5 y + 4 = (x + 3) squared + one-half?
y = one-fifth x squared + six-fifths x + eleven-tenths
y = 3 plus-or-minus StartRoot 5 x + seven-halves EndRoot
Negative 5 y minus 4 = negative (x + 3) squared minus one-half
y = negative 3 plus-or-minus StartRoot 5 x + seven-halves EndRoot

To find the inverse of the equation \( 5y + 4 = (x + 3)^2 + \frac{1}{2} \), we need to follow these steps:

### Step 1: Start with the given equation:
\[
5y + 4 = (x + 3)^2 + \frac{1}{2}
\]

### Step 2: Solve for \( y \):
\[
5y = (x + 3)^2 + \frac{1}{2} - 4
\]
\[
5y = (x + 3)^2 - \frac{7}{2}
\]
\[
y = \frac{(x + 3)^2 - \frac{7}{2}}{5}
\]

Now that we have \( y \) in terms of \( x \), we swap \( y \) and \( x \) to find the inverse:

\[
x = \frac{(y + 3)^2 - \frac{7}{2}}{5}
\]

### Step 3: Multiply both sides by 5:
\[
5x = (y + 3)^2 - \frac{7}{2}
\]

### Step 4: Add \( \frac{7}{2} \) to both sides:
\[
5x + \frac{7}{2} = (y + 3)^2
\]

### Step 5: Take the square root of both sides:
\[
y + 3 = \pm \sqrt{5x + \frac{7}{2}}
\]

### Step 6: Solve for \( y \):
\[
y = -3 \pm \sqrt{5x + \frac{7}{2}}
\]

Thus, the inverse of the original equation is:
\[
y = -3 \pm \sqrt{5x + \frac{7}{2}}
\]

The correct answer from the given choices is:
\[
\boxed{y = -3 \pm \sqrt{5x + \frac{7}{2}}}
\]

Would you like more details on how the square root was applied or any other step?

Here are 5 related questions for further exploration:
1. What is the domain of the inverse function?
2. How would you check if two functions are inverses?
3. How can you graph the inverse function of \( 5y + 4 = (x + 3)^2 + \frac{1}{2} \)?
4. What is the range of the original function?
5. How do you find the inverse of a cubic function?

**Tip:** When taking the square root while finding an inverse, always remember to include both the positive and negative roots (the "plus-minus" symbol).

Which equation is the inverse of 5y + 4 = (x + 3)^2 + 1/2? y = 1/5 x^2 + 6/5 x + 11/10 y = 3 ± √(5x + 7/2) -5y - 4 = -(x + 3)^2 - 1/2 y = -3 ± √(5x + 7/2)

Learn how to find the inverse of the quadratic equation 5y + 4 = (x + 3)^2 + 1/2 with a detailed step-by-step solution. This problem covers key algebraic concepts like inverse functions and square roots. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation