{"id":31822,"date":"2019-12-21T17:40:34","date_gmt":"2019-12-21T22:40:34","guid":{"rendered":"https:\/\/www.myedme.com\/login\/?p=31822"},"modified":"2019-12-22T09:02:04","modified_gmt":"2019-12-22T14:02:04","slug":"finding-an-inverse-function","status":"publish","type":"post","link":"https:\/\/myedme.com\/login\/finding-an-inverse-function\/","title":{"rendered":"Finding an Inverse Function"},"content":{"rendered":"\n

Functions take x<\/em> values to y<\/em> values. Inverse functions reverse the direction. Now y<\/em> values go to x<\/em> values. <\/p>\n\n\n\n

Some equations are easy to find the inverse of others because it is easy to see how to reverse the function for all values. <\/p>\n\n\n\n

Here are 3 examples of increasing complexity:<\/p>\n\n\n\n

  1. If f<\/em>(x<\/em>) = x<\/em> + 3, then we can reverse the function by solving x <\/em>= f<\/em>(x<\/em>) + 3 for f<\/em>(x<\/em>), which is f<\/em>(x<\/em>) = x<\/em> – 3. <\/li><\/ol>\n\n\n\n

    <\/p>\n\n\n\n

    2. If f<\/em>(x<\/em>) = 5x<\/em>, then we can reverse the function by solving x<\/em> = 5 f<\/em>(x<\/em>) for f<\/em>(x<\/em>), which is \"<\/p>\n\n\n\n

    3. If f<\/em>(x<\/em>)=x<\/em>3<\/sup>, then we can reverse the function by solving for x<\/em> = (f<\/em>(x<\/em>))3<\/sup> which is <\/p>\n\n\n\n\"\n\n\n\n

    <\/p>\n\n\n\n

    Same Content from OpenStax<\/a><\/h3>\n\n\n\n

    We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f<\/em> to elements in the range of f<\/em>. The inverse function maps each element from the range of f<\/em> back to its corresponding element from the domain of f<\/em>. Therefore, to find the inverse function of a one-to-one function f<\/em>, given any y<\/em> in the range of f<\/em>, we need to determine which x<\/em> in the domain of f<\/em> satisfies f<\/em>(x<\/em>)=y<\/em>. Since f<\/em> is one-to-one, there is exactly one such value x<\/em>. We can find that value x<\/em> by solving the equation f<\/em>(x<\/em>)=y<\/em> for x<\/em>. Doing so, we are able to write x<\/em> as a function of y<\/em> where the domain of this function is the range of f<\/em> and the range of this new function is the domain of f<\/em>. Consequently, this function is the inverse of f<\/em>, and we write x<\/em>=f<\/em>\u22121<\/sup>(y<\/em>). Since we typically use the variable x<\/em> to denote the independent variable and y<\/em> to denote the dependent variable, we often interchange the roles of x<\/em> and y<\/em>, and write y<\/em>=f<\/em>\u22121<\/sup>(x<\/em>). Representing the inverse function in this way is also helpful later when we graph a function f<\/em> and its inverse f<\/em>\u22121<\/sup> on the same axes.<\/p>\n\n\n\n

    PROBLEM-SOLVING STRATEGY: FINDING AN INVERSE FUNCTION<\/p>\n\n\n\n

    1. Solve the equation\u00a0y<\/em>=f<\/em>(x<\/em>)\u00a0for\u00a0x<\/em>.<\/li>
    2. Interchange the variables\u00a0x<\/em>\u00a0and\u00a0y<\/em>\u00a0and write\u00a0y<\/em>=f<\/em>\u22121(x<\/em>).<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

      Functions take x values to y values. Inverse functions reverse the direction. Now y values go to x values. Some equations are easy to find the inverse of others because it is easy to see how to reverse the function for all values. Here are 3 examples of increasing complexity: If f(x) = x + […]<\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-31822","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/31822","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/comments?post=31822"}],"version-history":[{"count":4,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/31822\/revisions"}],"predecessor-version":[{"id":31827,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/posts\/31822\/revisions\/31827"}],"wp:attachment":[{"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/media?parent=31822"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/categories?post=31822"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/myedme.com\/login\/wp-json\/wp\/v2\/tags?post=31822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}