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Exercises 1 Let's simplify the left-hand side of the given expression. The exponent we get after simplifying will be our answer. x-2⋅x0x5/3⋅x-1⋅3x​​na​=a1/nx-2⋅x0x5/3⋅x-1⋅x1/3​ Simplify numerator am⋅an=am+nx-2⋅x0x5/3−1+1/3​1=aa​x-2⋅x0x5/3−3/3+1/3​Add and subtract fractionsx-2⋅x0x3/3​aa​=1 x-2⋅x0x1​ Simplify denominator am⋅an=am+nx-2+0x1​Add terms x-2x1​ Simplify anam​=am−nx1−(-2)a−(-b)=a+bx1+2Add terms x3
Exercises 2 We are asked to find f(-7). Let's analyze the graph of f.First, we will identify the equation of the graph. We know that the function f is an exponential function. y=a⋅bx​ Notice that the points (0,1) and (-2,4) belong to the graph. We will find values of a and b by substituting the coordinates of these points into the equation. y=a⋅bxx=0, y=11=a⋅b0a0=11=a⋅1a⋅1=a1=a Let's substitute 1 for a into the equation. y=a⋅bx⇓y=1⋅bx⇓y=bx​ Next, we will substitute (-2,4) for x and y to find b. y=bxx=-2, y=44=b-2 Solve for b LHS-21​=RHS-21​4-21​=(b-2)-21​(am)n=am⋅n4-21​=b1a1=a4-21​=ba-m=am1​421​1​=bna​=an1​4​1​=bCalculate root21​=bRearrange equation b=21​ Let's substitute 21​ for b into the equation. y=bx⇓y=(21​)x​ Therefore, the function is f(x)=(21​)x. Finally, we will find f(-7) by substituting -7 for x. f(x)=(21​)xx=-7f(-7)=(21​)-7 Simplify right-hand side (ba​)m=bmam​f(-7)=2-71-7​1a=1f(-7)=2-71​am1​=a-mf(-7)=27Calculate power f(-7)=128
Exercises 3
Exercises 4 Let's analyze the given system of inequalities. Inequality 1y−2x≤4Inequality 26x−3y<​-12​​ We are asked to fill <​ with <, ≤, >, or ≥, so that the given system has no solution. Therefore, the solutions of each inequality should be completely different. Notice that the coefficients in both inequalities are proportional. Let's multiply Inequality 1 by -3. Inequality 1y−2x≤4Inequality 26x−3y<​-12​⇓Inequality 16x−3y≥-12Inequality 26x−3y<​-12​​ Notice that the both inequalities looks almost the same. Since we want the system to have no solutions, the signs should be opposite. Therefore, we should fill <​ with <.
Exercises 5 We know that the second term of the given sequence is 7. Furthermore, each term of the sequence is 10 more than the preceding term. Therefore the sequence is an arithmetic sequence, and its common difference is equal to d=10. So, we can fill one 00​ already. a1​=- 3​,an​=an−1​+10​an​=10​n−13​​ Since the second term is 7 and it is 10 more than the first term, we get that the first term is 10 less than 7. This is 7−10=-3. So, we can fill the space next to a1​. a1​=- 3​,an​=an−1​+10​an​=10​n−13​​ Next, we will find the explicit rule of the arithmetic sequence. an​=a1​+(n−1)d​ In our case, a1​=-3 and d=10. Let's substitute these values and find the rule! an​=a1​+(n−1)da1​=-3, d=10an​=-3+(n−1)(10) Simplify right-hand side Distribute 10an​=-3+n(10)−1(10)Multiplyan​=-3+10n−10Subtract terms an​=10n−13 Finally, we can fill all the spaces! a1​=- 3​,an​=an−1​+10​an​=10​n−13​​
Exercises 6 Let's start by graphing the function y=870−14.8t, which is a linear function written in the slope-intercept form. If you need explanation on graphing a linear function in slope-intercept form, please refer to here.window.JXQ = window.JXQ || []; window.JXQtable = window.JXQtable || {}; if(!window.JXQtable["Solution19836_0_1388782341_l"]) { window.JXQ.push( function () { var code = function (gid) { var images = []; try { var b=mlg.board([-175,1150,175,-150],{"grid":{"visible":false},"desktopSize":"medium",style:'usa',yScale:"5"}); b.xaxis(100,3,{name:"t"}); b.yaxis(400,3); var p1 = b.point(0,870); var p2 = b.point(30,870-14.8*30); b.slope(p1,p2,{fillColor:mlg.latexgreen,label1:{distance:45,position:0.5,name:'1'},label2:{distance:20,position:0.5,name:'14.8',rotateText:false}}); var func1 = b.func("870-14.8*x"); b.legend(func1,[100,400],"y=870-14.8t"); b.flag(p1,'y\\text{-intercept}',140,100);} catch( e ) { mw.hook("ml.error.report").fire(e, { tags: { module: "jsxg", graph: "Solution19836_0_1388782341_l" } }); console.error(e); } if(b) {b.board.update()} }; mw.loader.using('ext.MLJSXGraph', function() { window.ml.jsxgraph.buildRealGraph("Solution19836_0_1388782341_l", "Solution19836_0_1388782341_p", 1, code); }); } ); } window.JXQtable["Solution19836_0_1388782341_l"] = true;Here y is the height in feet of a hot-air balloon t minutes after it begins its descent. The y-intercept represents the initial height of 870 ft. The slope is the rate at which it descends, at 14.8 ft/min. This interpretation corresponds to Option B.
Exercises 7 We are asked to find all the functions whose x-value is an integer when f(x)=10. Let's consider the first function, f(x)=3x−2, and solve the equation f(x)=10 for x. f(x)=10f(x)=3x−23x−2=10 Solve for x LHS+2=RHS+23x=12LHS/3=RHS/3 x=4 Since the solution x=4 is an integer, the considered function should be included in the answer. Let's follow the same process for the other given functions!Function f(x)Equation f(x)=10SolutionIs It an Integer? f(x)=-2x+4-2x+4=10x=-3Yes ✓ f(x)=23​x+423​x+4=10x=4Yes ✓ f(x)=-3x+5-3x+5=10x=-35​No × f(x)=21​x−621​x−6=10x=32Yes ✓ f(x)=4x+144x+14=10x=-1Yes ✓
Exercises 8 Let's recall the standard forms of exponential growth and decay functions, and that of a linear function.An exponential decay function is in the form f(x)=a⋅bx, where a>0 and 0<b<1 or when a<0 and b>1. An exponential growth function is in the form f(x)=a⋅bx, where a>0 and b>1 or when a<0 and 0<b<1. A linear function is in the form f(x)=mx+b.Next, we will place each function into the right category.FunctionCategory f(x)=-2(8)xExponential Decay f(x)=15−x=-1x+15Linear f(x)=21​(3)xExponential Growth f(x)=6x2+9Neither f(x)=4(1.6)x/10Exponential Growth f(x)=x(18−x)=18x−x2Neither f(x)=3(61​)xExponential Decay f(x)=-3(4x+1−x)=-9x−3Linear f(x)=416​+2x=2x+416​Linear
Exercises 9 We want to start by drawing a graph of the given exponential function. f(x)=2x​ Because the base of the function is greater than 1, we know that this is an exponential growth function. Let's make a table of values.x2xy=2x -22-20.25 -12-10.5 0201 1212 2224 The ordered pairs (-2,0.25), (-1,0.5), (0,1), (1,2), and (2,4) all lie on the function. Now, we will plot and connect these points with a smooth curve.Comparison to the Shown Graph We are asked to compare the shown graph to the graph of f(x). Let's draw them together!Notice that the graph of f(x) should be vertically shrink by a factor of 2 and vertically shifted 3 units upward to get the given graph.