/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Suppose \(h(x)=x^{2}+3 x+4,\) wi... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 49

Suppose \(h(x)=x^{2}+3 x+4,\) with the domain of \(h\) being the set of positive numbers. Evaluate \(h^{-1}(7)\).

Short Answer

Expert verified
The short answer is: \(h^{-1}(7) = \frac{-3 + \sqrt{21}}{2}\).

Step by step solution

01

Finding the inverse function

We have the function \(h(x) = x^2 + 3x + 4\). Let the inverse function be \(h^{-1}(x) = y\). We can rewrite the function \(h(x)\) as: $$ x = y^2 + 3y + 4 $$ Now, we need to solve for \(y\) in terms of \(x\).
02

Solving for the inverse function

To solve for \(y\) in terms of \(x\), we can use the quadratic formula: $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ In our equation, \(a = 1\), \(b = 3\), and \(c = 4-x\): $$ y = \frac{-3 \pm \sqrt{3^2 - 4(1)(4-x)}}{2(1)} $$ Simplify the equation: $$ y = \frac{-3 \pm \sqrt{9 - 16 + 4x}}{2} $$ Now, since the domain of \(h(x)\) is the set of positive numbers, we will choose the positive square root so that \(h^{-1}(x)\) will also have a positive domain: $$ h^{-1}(x) = \frac{-3 + \sqrt{9 - 16 + 4x}}{2} $$
03

Evaluate \(h^{-1}(7)\)

Now that we've got the inverse function, we can plug in \(x = 7\) to evaluate \(h^{-1}(7)\): $$ h^{-1}(7) = \frac{-3 + \sqrt{9 - 16 + 4(7)}}{2} $$ Solve the equation: $$ h^{-1}(7) = \frac{-3 + \sqrt{9 - 16 + 28}}{2} $$ $$ h^{-1}(7) = \frac{-3 + \sqrt{21}}{2} $$ So, the value of \(h^{-1}(7)\) is \(\frac{-3 + \sqrt{21}}{2}\).

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