inverse of f(x)=log_{2}(x)+1

{ "query": { "display": "inverse $$f\\left(x\\right)=\\log_{2}\\left(x\\right)+1$$", "symbolab_question": "FUNCTION#inverse f(x)=\\log_{2}(x)+1" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "2^{x-1}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$\\log_{2}\\left(x\\right)+1:{\\quad}2^{x-1}$$", "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=\\log_{2}\\left(x\\right)+1" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=\\log_{2}\\left(x\\right)+1", "result": "x=\\log_{2}\\left(y\\right)+1", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=\\log_{2}\\left(y\\right)+1" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZecyuqOfecKK228+O2VPXHe+ghm46YmuFbmmG78CQ7UkuT7NM3IgGSg7KvASoIT1+CGdMQWj7euUj19HIJMZ9JN6PZq4pYr+MqS+U8UJjX531Kuuv9CNQvrdvkZ5dLi5r5KKTpUQqDZ50xP6ow8quMY=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=\\log_{2}\\left(y\\right)+1$$", "input": "x=\\log_{2}\\left(y\\right)+1", "steps": [ { "type": "step", "primary": "Switch sides", "result": "\\log_{2}\\left(y\\right)+1=x" }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\log_{2}\\left(y\\right)+1=x", "result": "\\log_{2}\\left(y\\right)=x-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "\\log_{2}\\left(y\\right)+1-1=x-1" }, { "type": "step", "primary": "Simplify", "result": "\\log_{2}\\left(y\\right)=x-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply log rules", "input": "\\log_{2}\\left(y\\right)=x-1", "result": "y=2^{x-1}", "steps": [ { "type": "step", "primary": "Use the logarithmic definition: If $$\\log_a\\left(b\\right)=c\\:$$then $$b=a^c$$", "secondary": [ "$$\\log_{2}\\left(y\\right)=x-1\\quad\\:\\Rightarrow\\:\\quad\\:y=2^{x-1}$$" ], "result": "y=2^{x-1}" } ], "meta": { "interimType": "Apply Log Rules Title 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7M87zTha4r0Lcgm43of+bz7lBEc5Lq3B+ElZNIXxsYSUdkJOvZf8WIsHralKZkogurcWKnNRE1CoAM/KEoNuMJU/9dCqx+Ahf6WYgPcWBNGsrDIrQoh956AWP+rNh/qnZBLgzlWkLyqH6h6GOYzWqbFSO8Um8HlrWMcP05MOaiuI=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "2^{x-1}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\log_{2}(x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }