16. Q16
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Similar Question 1
<p>Suppose that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> are functions.</p><p>We say that the real number <code class='latex inline'>c</code> is a real fixed point of <code class='latex inline'>f</code> if <code class='latex inline'>f (c) = c</code>.</p><p>We say that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> <em>commute</em> it <code class='latex inline'>f (g(x)) = g(f(x))</code> for all real numbers <code class='latex inline'>x</code>.</p><p>If <code class='latex inline'>f(x) = x^2 - 2</code>, determine all real fixed points of <code class='latex inline'>f</code>.</p>
Similar Question 2
<p>Suppose that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> are functions.</p><p>We say that the real number <code class='latex inline'>c</code> is a real fixed point of <code class='latex inline'>f</code> if <code class='latex inline'>f (c) = c</code>.</p><p>We say that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> <em>commute</em> it <code class='latex inline'>f (g(x)) = g(f(x))</code> for all real numbers <code class='latex inline'>x</code>.</p><p>If <code class='latex inline'>f(x) = x^2 - 2</code>, determine all real fixed points of <code class='latex inline'>f</code>.</p>
Similar Question 3
<img src="/qimages/13524" />
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Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Suppose that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> are functions.</p><p>We say that the real number <code class='latex inline'>c</code> is a real fixed point of <code class='latex inline'>f</code> if <code class='latex inline'>f (c) = c</code>.</p><p>We say that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> <em>commute</em> it <code class='latex inline'>f (g(x)) = g(f(x))</code> for all real numbers <code class='latex inline'>x</code>.</p><p>If <code class='latex inline'>f(x) = x^2 - 2</code>, determine all real fixed points of <code class='latex inline'>f</code>.</p>
<img src="/qimages/13524" />
<p>If <code class='latex inline'>f_{0}(x) = \dfrac{x}{x + 1}</code> and <code class='latex inline'>f_{n + 1} = f_{0} \circ f_{n}</code> for <code class='latex inline'>n = 0, 1, 2, ... </code>, find a formula for <code class='latex inline'>f_{n}(x)</code>.</p><p>Let me start you off... Note that <code class='latex inline'>f_{n + 1} = f_{0} \circ f_{n}</code> is a recursive formula. We&#39;ll get into this more when in the sequences and series.</p><p><code class='latex inline'> \begin{array}{lllll} f_{1} = f_{0}\circ f_{0} = f_{0}(\frac{x}{x+ 1}) = \dfrac{\frac{x}{x + 1}}{\frac{x}{x + 1} + 1} \times \dfrac{x + 1}{x + 1} = \dfrac{x}{x + (x + 1)} = \dfrac{x}{2x + 1} \end{array} </code></p>
<p>If <code class='latex inline'>f(x) = x^2 - 2x</code>, find the sum of all values of <code class='latex inline'>x</code> for which <code class='latex inline'>f(x) = f[f(x)]</code>.</p>
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