intercepts of f(x)=2x-4sin(x),0<= x<= 2pi

{ "query": { "display": "intercepts $$f\\left(x\\right)=2x-4\\sin\\left(x\\right),\\:0\\le\\:x\\le\\:2π$$", "symbolab_question": "FUNCTION#intercepts f(x)=2x-4\\sin(x),0\\le x\\le 2π" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{Y\\:Intercepts}: (0,0)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$2x-4\\sin\\left(x\\right),\\:0\\le\\:x\\le\\:2π:\\quad\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$y-$$axis interception point of $$2x-4\\sin\\left(x\\right),\\:0\\le\\:x\\le\\:2π:{\\quad}\\left(0,\\:0\\right)$$", "input": "2x-4\\sin\\left(x\\right),\\:0\\le\\:x\\le\\:2π", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=2\\cdot\\:0-4\\sin\\left(0\\right):{\\quad}0$$", "input": "2\\cdot\\:0-4\\sin\\left(0\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=2\\cdot\\:0-4\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfnw2qj3gmJTN30ZmoCVZnF32Ye59ANYl0qWTO1f6KL83+qIgQZ8nxuQmI14y/0RT6WllV+scyoPpqg9oK3OTtea6EyITWinhZrzbnu6JO/WS9c++/gGdJ3AWVjctLnPtgxOz7E3+Mgy/W7WqlLeYM6gJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\left\\{2x-4\\sin(x):0\\le x\\le 2π\\right\\}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }