Questions: Find the inverse of the function. y=-(5x+4)^2-7 Write your answer in the form a ± b sqrt(cx+d), where a, b, c, and d are constants. Simplify any fractions. y=

$Find the inverse of the function. y=-(5x+4)^2-7 Write your answer in the form a ± b sqrt(cx+d), where a, b, c, and d are constants. Simplify any fractions. y=$

Transcript text: Find the inverse of the function. \[ y=-(5 x+4)^{2}-7 \] Write your answer in the form $\mathrm{a} \pm \mathrm{b} \sqrt{\mathrm{cx}+\mathrm{d}}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$, and d are constants. Simplify any fractions. \[ y= \]

Solution

Solution Steps

To find the inverse of the given quadratic function, we need to follow these steps:

Replace \( y \) with \( x \) and \( x \) with \( y \) in the equation.
Solve the resulting equation for \( y \).
Express the solution in the form \( a \pm b \sqrt{cx + d} \).

Step 1: Original Function

The given quadratic function is

\[ y = -(5x + 4)^2 - 7. \]

Step 2: Replace Variables

To find the inverse, we replace \( y \) with \( x \) and \( x \) with \( y \):

\[ x = -(5y + 4)^2 - 7. \]

Step 3: Rearranging the Equation

Rearranging the equation gives us:

\[ x + 7 = -(5y + 4)^2. \]

Step 4: Solving for \( y \)

Next, we solve for \( y \):

\[ -(5y + 4)^2 = x + 7 \implies (5y + 4)^2 = - (x + 7). \]

Taking the square root of both sides, we have:

\[ 5y + 4 = \pm \sqrt{- (x + 7)}. \]

Step 5: Isolating \( y \)

Now, isolating \( y \):

\[ 5y = -4 \pm \sqrt{- (x + 7)} \implies y = -\frac{4}{5} \pm \frac{1}{5} \sqrt{- (x + 7)}. \]

Final Answer

Thus, the inverse function can be expressed as:

\[ y = -\frac{4}{5} \pm \frac{1}{5} \sqrt{- (x + 7)}. \]

The final answer is

\[ \boxed{y = -\frac{4}{5} \pm \frac{1}{5} \sqrt{- (x + 7)}} \]

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