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Similar Question 1
<p>Repeat Question 17 for <code class='latex inline'>f^{-1}</code> and <code class='latex inline'>g^{-1}</code>.</p><p>(Question 17)</p><p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Compare the graphs of <code class='latex inline'>f</code> and <code class='latex inline'>g</code>. Describe how they are</p><p>(a) alike</p><p>(b) different</p>
Similar Question 2
<p>Write the equation of each inverse function below in </p><p><strong>i)</strong> exponential form</p><p><strong>ii)</strong> logarithmic form</p><p><code class='latex inline'> \displaystyle \begin{array}{llllll} &(a) & f(x) = 4^x &(c) & f(x) = (\frac{1}{3})^x\\ &&&&\\ &(c) & f(x) = 8^x &(d) & f(x) = (\frac{1}{5})^x\\ \end{array} </code></p>
Similar Question 3
<p>Which graph represents the inverse of the exponential function?</p><p><code class='latex inline'> \displaystyle f(x)= 4^x </code></p>
Similar Questions
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>Which graph represents the inverse of the exponential function?</p><p><code class='latex inline'> \displaystyle f(x)= 4^x </code></p>
<img src="/qimages/2275" /><p>(b) Write an equation for this exponential function.</p><p>(b) Sketch a graph of the inverse of the function by reflecting its graph in the line <code class='latex inline'>y=x</code></p>
<p>Which graph represents the inverse of the exponential function?</p><p><code class='latex inline'> \displaystyle f(x)= (\frac{1}{5})^x </code></p>
<p>Write the equation of the inverse of each exponential functions in logarithmic form.</p><p><code class='latex inline'> \displaystyle \begin{array}{lllll} &(a) & f(x) = 3^x &(c) & f(x) = (\frac{1}{4})^x\\ &&&&\\ &(c) & f(x) = 10^x &(d) & f(x) = m^x\\ \end{array} </code></p>
<p>For the function <code class='latex inline'>y=1.5^x</code> over the domain <code class='latex inline'>0\leq x\leq 6</code> </p><p>Estimate the instantaneous rate of change of <code class='latex inline'>y</code> with respect to <code class='latex inline'>x</code> at </p><p><code class='latex inline'>x =1, 2, 3, 4, 5</code></p>
<p>Write an equation for the inverse of the function shown: <code class='latex inline'> \displaystyle x = (\frac{1}{10})^y </code></p><img src="/qimages/2273" />
<p>Write the equation of each inverse function below in </p><p><strong>i)</strong> exponential form</p><p><strong>ii)</strong> logarithmic form</p><p><code class='latex inline'> \displaystyle \begin{array}{llllll} &(a) & f(x) = 4^x &(c) & f(x) = (\frac{1}{3})^x\\ &&&&\\ &(c) & f(x) = 8^x &(d) & f(x) = (\frac{1}{5})^x\\ \end{array} </code></p>
<p>Repeat Question 17 for <code class='latex inline'>f^{-1}</code> and <code class='latex inline'>g^{-1}</code>.</p><p>(Question 17)</p><p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Compare the graphs of <code class='latex inline'>f</code> and <code class='latex inline'>g</code>. Describe how they are</p><p>(a) alike</p><p>(b) different</p>
<p>An influenza virus is spreading through a school according to the function <code class='latex inline'>N=10(2)^t</code>, where <code class='latex inline'>N</code> is the number of people infected and <code class='latex inline'>t</code> is the number of people infected and <code class='latex inline'>t</code> is the time, in days.</p><p>(a) How many people have the virus at each time?</p> <ul> <li>i) initially, when <code class='latex inline'>t=0</code></li> <li>ii) after 1 day</li> <li>iii) after 2 days</li> <li>iv) after 3 days</li> </ul> <p>(b) Determine the average rate of change between day 1 and day 2.</p><p>(d) Estimate the instantaneous rate of change after</p> <ul> <li>i) 1 day </li> <li>ii) 2 days</li> </ul>
<p>Find the inverse of <code class='latex inline'>f</code> and <code class='latex inline'>g</code>.</p><p><code class='latex inline'>f(x)=4^x</code></p><p>Identify the key features of <code class='latex inline'>f</code></p><p><strong>(a)</strong> domain and range</p><p><strong>(b)</strong> <code class='latex inline'>x</code>-intercept, if it exists</p><p><strong>(c)</strong> <code class='latex inline'>y</code>-intercept, if it exists</p><p><strong>(d)</strong> intervals for which the function is positive and intervals for which it is negative</p><p><strong>(e)</strong> intervals for which the function is increasing and intervals for which it is decreasing</p><p><strong>(f)</strong> equation of the asymptote</p>
<p>Match <code class='latex inline'>a, b, c, d</code> with inverse of <code class='latex inline'>i, ii, iii, iv</code></p><p>(a) <img src="/qimages/2264" /></p><p>(b) <img src="/qimages/2265" /></p><p>(c) <img src="/qimages/2266" /></p><p>(d) <img src="/qimages/2267" /></p><p>i) <code class='latex inline'>y=5^x</code></p><p>ii) <code class='latex inline'>\displaystyle{\left(\frac{1}{2}\right)^x}</code></p><p>iii) <code class='latex inline'>y=2^x</code></p><p>iv) <code class='latex inline'>\displaystyle{y=\left(\frac{1}{5}\right)^x}</code></p>
<p>For the function <code class='latex inline'>y=1.5^x</code> over the domain <code class='latex inline'>0\leq x\leq 6</code> </p><p>determine the average rate of change of <code class='latex inline'>y</code> with respect to <code class='latex inline'>x</code> for each interval.</p><p><em>i)</em> <code class='latex inline'>x=1</code> to <code class='latex inline'>x=2</code></p><p><em>ii)</em> <code class='latex inline'>x=2</code> to <code class='latex inline'>x=3</code></p><p><em>iii)</em> <code class='latex inline'>x=3</code> to <code class='latex inline'>x=4</code></p><p><em>iv)</em> <code class='latex inline'>x=4</code> to <code class='latex inline'>x=5</code></p>
<p>Explain how you can use the graph of <code class='latex inline'>y =\log_2x</code> (below) to help you determine the solution to <code class='latex inline'>2^y=8</code></p><img src="/qimages/689" />
<p>Which graph represents the inverse of the exponential function?</p><p><code class='latex inline'> \displaystyle f(x)= (\frac{1}{3})^x </code></p>
<p>A spaceship approaches Planet X and the planet’s force of gravity starts to pull the ship in. To prevent a crash, the crew must engage the thrusters when the ship is exactly 100 km from the planet. The distance away from the planet can be modelled by the function <code class='latex inline'>d(t)=(1.4)^t</code>, where <code class='latex inline'>d</code> represents the distance, in hundreds of kilometres, between the ship and the planet and <code class='latex inline'>t</code> represents the time, in seconds.</p><p>(a) What is the ship’s average velocity between 1 s and 2 s? between 3 s and 4 s?</p><p>(b) What is the ship’s instantaneous velocity at 3 s? at 4 s?</p>
<img src="/qimages/2276" /><p>(a) Write an equation for this exponential function.</p><p>(c) Sketch a graph of the inverse of the function by reflecting its graph in the line <code class='latex inline'>y=x</code></p>
<p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Compare the graphs of <code class='latex inline'>f</code> and <code class='latex inline'>g</code>. Describe how they are</p><p>(a) alike</p><p>(b) different</p>
<p>Match each equation to its corresponding graph.</p><p>i) <code class='latex inline'>y=5^x</code></p><p>ii) <code class='latex inline'>\displaystyle{\left(\frac{1}{2}\right)^x}</code></p><p>iii) <code class='latex inline'>y=2^x</code></p><p>iv) <code class='latex inline'>\displaystyle{y=\left(\frac{1}{5}\right)^x}</code></p><p>(a) <img src="/qimages/2268" /></p><p>(b) <img src="/qimages/2269" /></p><p>(c) <img src="/qimages/2270" /></p><p>(d) <img src="/qimages/2271" /></p>
<p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Identify the key features of <code class='latex inline'>g</code>.</p><p>(a) domain and range</p><p>(b) <code class='latex inline'>x</code>-intercept, if it exists</p><p>(c) <code class='latex inline'>y</code>-intercept, if it exists</p><p>(d) intervals for which the function is positive and intervals for which it is negative</p><p>(e) intervals for which the function is increasing and intervals for which it is decreasing</p><p>(f) equation of the asymptote</p>
<img src="/qimages/2272" /><p>a) </p> <ul> <li>Graph the line <code class='latex inline'>y=x</code> on the same grid.</li> <li>Graph the inverse of this function by reflecting it in the line <code class='latex inline'>y=x</code>.</li> </ul> <p>b) Write an equation for the inverse function.</p>
<p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Identify the key features of <code class='latex inline'>g</code>.</p><p>(a) domain and range</p><p>(b)<code class='latex inline'>x</code>-intercept, if it exists</p><p>(c) <code class='latex inline'>y</code>-intercept, if it exists</p><p>(d) intervals for which the function is positive and intervals for which it is negative</p><p>(e) intervals for which the function is increasing and intervals for which it is decreasing</p><p>(f) equation of the asymptote</p>
<p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p> <ul> <li><p>Sketch a graph of <code class='latex inline'>f</code></p></li> <li><p>Graph the line <code class='latex inline'>y=x</code> on the same grid.</p></li> <li><p>Sketch the inverse of <code class='latex inline'>f</code> on the same grid by reflecting <code class='latex inline'>f</code> in the line <code class='latex inline'>y=x</code>.</p></li> </ul> <p>Which of the following is the final picture of the conditions above.</p>
<p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p> <ul> <li>Sketch a graph of the function <code class='latex inline'>g</code></li> <li>Graph the line <code class='latex inline'>y=x</code> on the same grid.</li> <li>Sketch the inverse of g on the same grid by reflecting g in the line <code class='latex inline'>y=x</code>.</li> </ul> <p>Which of the following is the final image of the three steps combined above?</p>
<p>(a) For the graph of any function <code class='latex inline'>f(x)=b^x</code> and its inverse, describe the points where the <code class='latex inline'>x</code>—coordinates and <code class='latex inline'>y</code>-coordinates are equal. Explain how the functions relate to the line <code class='latex inline'>y=x</code> at these points.</p><p>(b) How does the result differ when <code class='latex inline'>b>1</code> versus when <code class='latex inline'>0<b<1</code>?</p>
<p>For <code class='latex inline'>f(x) = 4^{x-3}</code> find</p> <ol> <li>symmetry, if any</li> <li>x intercept, if any</li> <li>y intercept, if any</li> <li>asymptotes, if any</li> <li>transformation mapping</li> </ol>
<p><code class='latex inline'>f(x)=4^x</code></p><p><code class='latex inline'>\displaystyle{g(x)=\left(\frac{1}{2}\right)^x}</code></p><p>Identify the key features of <code class='latex inline'>f</code></p><p><strong>(a)</strong> domain and range</p><p><strong>(b)</strong> <code class='latex inline'>x</code>-intercept, if it exists</p><p><strong>(c)</strong> <code class='latex inline'>y</code>-intercept, if it exists</p><p><strong>(d)</strong> intervals for which the function is positive and intervals for which it is negative</p><p><strong>(e)</strong> intervals for which the function is increasing and intervals for which it is decreasing</p><p><strong>(f)</strong> equation of the asymptote</p>
<p>Which graph represents the inverse of the exponential function?</p><p><code class='latex inline'> \displaystyle f(x)= 8^x </code></p>
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