y=x

{ "query": { "display": "$$y=x$$", "symbolab_question": "FUNCTION#y=x" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{X\\:Intercepts}: (0,0),\\mathrm{Y\\:Intercepts}: (0,0)<br/>\\mathrm{Slope}: m=1<br/>\\mathrm{Inverse}: x<br/>\\mathrm{Domain}: -\\infty <x<\\infty <br/>\\mathrm{Range}: -\\infty <f(x)<\\infty ", "interval": "\\mathrm{Domain}: (-\\infty ,\\infty )<br/>\\mathrm{Range}: (-\\infty ,\\infty )" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Axis interception points of $$x:\\quad\\:$$X Intercepts$$:\\:\\left(0,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$x:{\\quad}\\left(0,\\:0\\right)$$", "input": "x", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$x=0:{\\quad}x=0$$", "input": "x=0", "steps": [ { "type": "step", "primary": "Could not simplify further", "result": "x=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7O66nweR+nNz9H3W2sUzL2BDX0k+EM/0vwN/YfwBX0du5KbKNRP2lUnpKreVmLXLgy+iO15zBWGX5plAt+BVIbj" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$x:{\\quad}\\left(0,\\:0\\right)$$", "input": "x", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "step", "result": "y=0" }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfmQ1Xa7hVHArl3wlOlTyCDjDm0ZOvA+Ygb668PxLRGXFzczxXhjhAZFo7dyFFV2xFTfEqUseV9ISoqDnjbA0w9a" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(0,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Slope of $$x:{\\quad}m=1$$", "input": "x", "steps": [ { "type": "step", "primary": "For a line equation for the form of $$\\mathbf{y=mx+b}$$, the slope is $$\\mathbf{m}$$", "result": "m=1" } ], "meta": { "interimType": "Slope Equation Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMGyxztx1DIZ4Y9QoeLjXWS/CK3p1wow19A9GBzcRtecKLp5UlQB3toMtUsvh0BgG9UZM339tm9U+faYzmhUe75w==" } }, { "type": "interim", "title": "Inverse of $$x:{\\quad}x$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=x" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=x", "result": "x=y", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=y" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZTCQuoId+lmzGwSGOsVB9tAZCZhUhxmkivATKpP0B+nz/CPEZTJfFYHFGpcyrRtI6k3kCh3oevUunZ7/b0qFKBSaOqV7KC2VKzOh7MoOPPaTsYnClkCALGbhY+VxTjGPmQ==" } }, { "type": "interim", "title": "Solve for $$y,\\:x=y$$", "input": "x=y", "steps": [ { "type": "step", "primary": "Switch sides", "result": "y=x" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "x" } ], "meta": { "solvingClass": "Function Inverse", "interimType": "Function Inverse Top 1Eq" } }, { "type": "interim", "title": "Domain of $$x\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$x:{\\quad}-\\infty\\:<f\\left(x\\right)<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "The range of polynomials with odd degree is all the real numbers", "result": "-\\infty\\:<f\\left(x\\right)<\\infty\\:" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Top 1Eq" } } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x" }, "showViewLarger": true } } }