{"id":2815,"date":"2026-10-11T08:00:00","date_gmt":"2026-10-11T02:30:00","guid":{"rendered":"https:\/\/digitoolkit.in\/blog\/?p=2815"},"modified":"2026-10-11T08:00:00","modified_gmt":"2026-10-11T02:30:00","slug":"function-formula-explained-linear-quadratic-composite-examples","status":"publish","type":"post","link":"https:\/\/digitoolkit.in\/blog\/function-formula-explained-linear-quadratic-composite-examples\/","title":{"rendered":"Function Formula Explained with Examples (Linear, Quadratic, Composite)"},"content":{"rendered":"<div style=\"background:#f2f7fb;border-left:4px solid #2271b1;padding:16px 20px;margin:0 0 24px;border-radius:4px;\">\n<p><strong>Quick Answer:<\/strong> Function formulas are rules that turn an input x into one output f(x). The main ones are linear (mx + c), quadratic (ax^2 + bx + c), composite g(f(x)) and inverse. This guide explains each with worked Indian examples in rupees, such as GST, taxi fares and profit.<\/p>\n<p><strong>Key takeaways:<\/strong><\/p>\n<ul>\n<li>The standard notation is y = f(x): x is the input and f(x) is the output.<\/li>\n<li>Linear functions use f(x) = mx + c; quadratics use f(x) = ax^2 + bx + c.<\/li>\n<li>Composite functions apply one rule first and then feed the result to a second rule.<\/li>\n<li>An inverse function reverses the rule; swap x and y, then solve.<\/li>\n<li>Always check the domain, because some inputs make a formula undefined.<\/li>\n<\/ul>\n<\/div>\n<h2>Function formulas: the short version<\/h2>\n<p>When students say &#8220;function formula&#8221;, they usually mean one of two things. Either they want the general notation, like y = f(x), or they want the formula for a specific type, like f(x) = mx + c for a straight line. Both ideas appear in the CBSE and ICSE syllabus, and both show up again in JEE and CUET papers.<\/p>\n<p>This article explains the main function formulas in plain words. You will see each formula, what every letter means, and worked examples using rupees, lakhs and Indian business cases. If you want to test any formula quickly, try the free <a href=\"https:\/\/digitoolkit.in\/calculators\/function-calculator\/\">function calculator<\/a> after reading.<\/p>\n<h2>The basic notation<\/h2>\n<p>The standard way to write a function is <code>y = f(x)<\/code>. Here x is the input, y is the output, and f is the name of the rule. You can use other letters. A profit rule might be P(x), and a cost rule might be C(n). The letter before the bracket is just a label.<\/p>\n<p>The NCERT Class 11 book defines a function as a special relation where every element of the first set is linked to exactly one element of the second set. In simple words, one input gives only one output. If you want the official wording, read the Relations and Functions chapter on ncert.nic.in.<\/p>\n<h3>Main function formulas at a glance<\/h3>\n<table>\n<thead>\n<tr>\n<th>Name<\/th>\n<th>Formula<\/th>\n<th>What the letters mean<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant<\/td>\n<td>f(x) = c<\/td>\n<td>The output is always the same number c<\/td>\n<\/tr>\n<tr>\n<td>Linear<\/td>\n<td>f(x) = mx + c<\/td>\n<td>m is the slope, c is the starting value<\/td>\n<\/tr>\n<tr>\n<td>Quadratic<\/td>\n<td>f(x) = ax^2 + bx + c<\/td>\n<td>a is not zero; the graph is a parabola<\/td>\n<\/tr>\n<tr>\n<td>Composite<\/td>\n<td>(g o f)(x) = g(f(x))<\/td>\n<td>Apply f first, then g<\/td>\n<\/tr>\n<tr>\n<td>Inverse<\/td>\n<td>f^-1(f(x)) = x<\/td>\n<td>Undoes what f did<\/td>\n<\/tr>\n<tr>\n<td>Exponential<\/td>\n<td>f(x) = a^x<\/td>\n<td>a is a positive number, not 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<blockquote>\n<p><strong>Key takeaway:<\/strong> Every function formula is only a recipe. The letters are placeholders, and your job is to replace them with real numbers in the right order.<\/p>\n<\/blockquote>\n<h2>The linear function formula<\/h2>\n<p>The linear function is f(x) = mx + c. The number m tells you how fast the output changes when x goes up by 1. The number c tells you where you start when x is 0. Mobile data plans, taxi fares and daily wages often follow this pattern.<\/p>\n<p>To find the slope from two points (x1, y1) and (x2, y2), use <code>m = (y2 - y1) \/ (x2 - x1)<\/code>. This is one of the most used formulas in coordinate geometry.<\/p>\n<h2>The quadratic function formula<\/h2>\n<p>A quadratic function is f(x) = ax^2 + bx + c, where a is not zero. Its graph is a U-shaped curve. If a is positive, the curve opens upward and has a lowest point. If a is negative, it opens downward and has a highest point. The turning point has x-value <code>x = -b \/ (2a)<\/code>.<\/p>\n<p>To find where the output becomes zero, use the quadratic formula: <code>x = (-b +\/- sqrt(b^2 - 4ac)) \/ (2a)<\/code>. The part under the root, b^2 &#8211; 4ac, is called the discriminant. Positive means two real roots, zero means one, and negative means none in real numbers.<\/p>\n<h2>Composite functions<\/h2>\n<p>A composite function joins two rules. You apply the first rule, then feed its answer into the second rule. The notation g(f(x)) means &#8220;do f first, then g&#8221;. The order matters in general, though not always, as the first example below shows.<\/p>\n<h2>Inverse functions<\/h2>\n<p>An inverse function reverses the rule. If f turns x into y, then the inverse turns y back into x. To find it, write y = f(x), swap x and y, and solve for y. Only functions that give different outputs for different inputs (one-one functions) have an inverse.<\/p>\n<h2>Worked example 1: Discount then GST (composite function)<\/h2>\n<p>A festive sale in Jaipur offers 20 percent off on a Rs 1,000 kurta. After the discount, 18 percent GST (used here as an example rate) is added.<\/p>\n<p>Let f(x) = 0.8x be the discount rule and g(x) = 1.18x be the GST rule.<\/p>\n<p><strong>Step 1:<\/strong> Apply the discount: f(1000) = 0.8 x 1000 = Rs 800.<br \/><strong>Step 2:<\/strong> Add GST: g(800) = 1.18 x 800 = Rs 944.<\/p>\n<p>As one formula, g(f(x)) = 1.18 x 0.8x = 0.944x. Check: 0.944 x 1000 = Rs 944. The customer pays Rs 944.<\/p>\n<p>Now reverse the order. Add GST first: 1.18 x 1000 = Rs 1,180. Then discount: 0.8 x 1180 = Rs 944. Same answer. This happens here because both steps are plain multiplications. With rules like f(x) = x &#8211; 100 and g(x) = 1.18x, the order would change the answer. For GST rates and rules, always refer to the official GST portal at gst.gov.in.<\/p>\n<h2>Worked example 2: Taxi fare and its inverse (linear function)<\/h2>\n<p>A cab company in Bengaluru charges a base fare of Rs 50 plus Rs 12 per km. The fare is f(x) = 12x + 50, where x is the distance in km.<\/p>\n<p><strong>Fare for 20 km:<\/strong> f(20) = 12(20) + 50 = 240 + 50 = Rs 290.<\/p>\n<p>Now suppose a rider paid Rs 410 and wants to know how far the trip was. We need the inverse function.<\/p>\n<p><strong>Step 1:<\/strong> Write y = 12x + 50.<br \/><strong>Step 2:<\/strong> Swap letters: x = 12y + 50.<br \/><strong>Step 3:<\/strong> Solve for y: 12y = x &#8211; 50, so y = (x &#8211; 50) \/ 12.<br \/><strong>Step 4:<\/strong> The inverse is f^-1(x) = (x &#8211; 50) \/ 12.<br \/><strong>Step 5:<\/strong> Put in Rs 410: (410 &#8211; 50) \/ 12 = 360 \/ 12 = 30.<\/p>\n<p><strong>Answer:<\/strong> The trip was 30 km. A quick check proves it: 12 x 30 + 50 = 410. The inverse is a clean way to turn a bill back into a distance.<\/p>\n<h2>Worked example 3: Profit of a small factory (quadratic function)<\/h2>\n<p>A small unit in Surat makes cotton shirts. If it makes x hundred shirts a month, its profit in thousands of rupees is P(x) = -x^2 + 60x &#8211; 500. Find the number of shirts that gives the highest profit, and the break-even points.<\/p>\n<p><strong>Highest profit:<\/strong> Here a = -1 and b = 60. The turning point is x = -b \/ (2a) = -60 \/ (-2) = 30. So 30 hundred shirts, which is 3,000 shirts.<\/p>\n<p><strong>Maximum profit:<\/strong> P(30) = -(30)^2 + 60(30) &#8211; 500 = -900 + 1800 &#8211; 500 = 400. That is Rs 400 thousand, or Rs 4 lakh a month.<\/p>\n<p><strong>Break-even points:<\/strong> Solve -x^2 + 60x &#8211; 500 = 0, which is the same as x^2 &#8211; 60x + 500 = 0. The discriminant is 60^2 &#8211; 4(1)(500) = 3600 &#8211; 2000 = 1600, and its square root is 40. So x = (60 &#8211; 40) \/ 2 = 10 or x = (60 + 40) \/ 2 = 50.<\/p>\n<p><strong>Answer:<\/strong> The unit makes a profit only when it produces between 1,000 and 5,000 shirts. The best output is 3,000 shirts, giving Rs 4 lakh. Business owners use this type of formula to plan production. Check numbers like these with a <a href=\"https:\/\/digitoolkit.in\/calculators\/quadratic-equation-calculator\/\">quadratic equation calculator<\/a> to avoid slips in the square root step.<\/p>\n<h2>Benefits of knowing the formulas<\/h2>\n<p>Knowing function formulas saves time in exams. You do not need to rebuild the idea every time, because you already know the pattern. It also helps you read graphs, since slope and turning points come straight from the formula. In business, the same formulas help you estimate cost, revenue and profit for different levels of sales. In personal finance, they help you compare plans, such as two loan offers with different fees. And once you understand the formulas, any software tool becomes easier to use, because you can tell if the output looks sensible.<\/p>\n<h2>Challenges and limitations<\/h2>\n<p>A formula only works inside its domain. The rule f(x) = 1 \/ x fails at x = 0, and the rule f(x) = sqrt(x) has no real value for negative x. Real life data is also messy. A profit curve like the one above is a neat model, but real costs rise in steps, prices change and demand moves. Use formulas as guides, not as promises.<\/p>\n<p>Another limit is memory. There are many formulas, and students often mix them. Writing them in one table, like the one above, and revising it weekly is a simple fix.<\/p>\n<h2>Common mistakes to avoid<\/h2>\n<ol>\n<li><strong>Mixing up g(f(x)) and f(g(x)).<\/strong> The function nearest to x is applied first. Read composite functions from the inside out.<\/li>\n<li><strong>Confusing the inverse with the reciprocal.<\/strong> The inverse f^-1(x) is not 1 \/ f(x). For f(x) = 2x, the inverse is x \/ 2, while the reciprocal is 1 \/ (2x).<\/li>\n<li><strong>Forgetting to swap x and y.<\/strong> When finding an inverse, swap the letters before you solve. Skipping this gives the original function back.<\/li>\n<li><strong>Using the quadratic formula with wrong signs.<\/strong> Write a, b and c with their signs first. If b is -60, then -b is +60.<\/li>\n<li><strong>Ignoring the domain.<\/strong> Always state the allowed inputs. A negative number of shirts makes no sense in a factory problem.<\/li>\n<li><strong>Assuming every function has an inverse.<\/strong> A function like f(x) = x^2 gives the same output for 3 and -3, so it has no inverse unless you limit the domain.<\/li>\n<\/ol>\n<h2>Best practices and expert recommendations<\/h2>\n<ol>\n<li><strong>Make a one-page formula sheet.<\/strong> Group linear, quadratic, composite and inverse formulas together, and review it before every test.<\/li>\n<li><strong>Name the type first.<\/strong> Before you calculate, decide whether the rule is linear, quadratic or something else. That choice tells you which formula to use.<\/li>\n<li><strong>Verify with a second input.<\/strong> After finding an inverse or a turning point, plug in one more number to confirm the result.<\/li>\n<li><strong>Use real contexts.<\/strong> Turn each formula into a story, such as a taxi fare or a shop discount, so that you remember its meaning, not just its shape.<\/li>\n<li><strong>Sketch a quick graph.<\/strong> Even a rough picture shows whether your answer is reasonable. A downward parabola should have a highest point, not a lowest one.<\/li>\n<li><strong>Practise from official material.<\/strong> NCERT books and past CBSE papers on cbse.gov.in give the exact style of question you will meet in school exams.<\/li>\n<\/ol>\n<h2>Try it yourself<\/h2>\n<p>Take the taxi rule f(x) = 12x + 50 and find the fare for 35 km by hand. Then check it using our free online function calculator. Try changing the base fare or the per-km rate to see how the answer moves. If a formula gives you an equation that you need to solve, an <a href=\"https:\/\/digitoolkit.in\/calculators\/equation-solver\/\">equation solver<\/a> is a useful next step.<\/p>\n<h2>Conclusion<\/h2>\n<p>Function formulas are simple once you see the pattern. The linear formula models steady change, the quadratic formula models peaks and break-even points, and composite and inverse functions let you chain or reverse rules. Learn what each letter means, practise with Indian examples, and always check your work with a second method or a trusted online tool.<\/p>\n<div data-dtk-related=\"1\" style=\"background:#f8f9fb;border:1px solid #e2e8f0;border-radius:6px;padding:16px 20px;margin:28px 0;\">\n<p style=\"margin:0 0 10px;\"><strong>Related tools &amp; guides on DigiToolkit<\/strong><\/p>\n<ul style=\"margin:0;padding-left:20px;\">\n<li><a href=\"https:\/\/digitoolkit.in\/calculators\/function-calculator\/\">Try the free Function Calculator &rarr;<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/how-to-calculate-function-value-step-by-step\/\">How to Calculate a Function Value Step by Step (With Indian Examples)<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/what-is-a-function-in-maths-simple-guide-for-students\/\">What Is a Function in Maths? A Simple Guide for Students<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/free-online-function-calculator-guide-how-to-use\/\">Function Calculator: Free Online Tool + Guide for Students<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/function-calculator-examples-for-beginners-with-answers\/\">Function Calculator Examples for Beginners (With Answers)<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/how-to-calculate-exponents-step-by-step\/\">How to Calculate Exponents Step by Step (With Indian Examples)<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/what-is-a-ratio-simple-guide\/\">What Is a Ratio? A Simple Guide for Beginners (India)<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/category\/mathematics\/\">More Mathematics guides<\/a><\/li>\n<\/ul>\n<\/div>\n<h2>FAQs<\/h2>\n<p><strong>What is the general formula of a function?<\/strong><br \/>The general form is y = f(x), where x is the input and f is the rule. The actual formula depends on the type, such as f(x) = mx + c for a linear function.<\/p>\n<p><strong>How do I find the inverse of a function?<\/strong><br \/>Write y = f(x), swap x and y, and then solve for y. For f(x) = 12x + 50, the inverse is (x &#8211; 50) \/ 12.<\/p>\n<p><strong>What is a composite function?<\/strong><br \/>A composite function applies one function and then another. The notation g(f(x)) means find f(x) first, then put that answer into g.<\/p>\n<p><strong>What is the turning point of a quadratic function?<\/strong><br \/>The turning point has x equal to -b \/ (2a). Put that x back into the function to get the highest or lowest value.<\/p>\n<p><strong>Do all functions have an inverse?<\/strong><br \/>No. Only one-one functions have an inverse. A function such as f(x) = x^2 has no inverse over all real numbers, because 3 and -3 give the same output.<\/p>\n<p><script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"What is the general formula of a function?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"The general form is y = f(x), where x is the input and f is the rule. The actual formula depends on the type, such as f(x) = mx + c for a linear function.\"}},{\"@type\":\"Question\",\"name\":\"How do I find the inverse of a function?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Write y = f(x), swap x and y, and then solve for y. For f(x) = 12x + 50, the inverse is (x - 50) \/ 12.\"}},{\"@type\":\"Question\",\"name\":\"What is a composite function?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"A composite function applies one function and then another. 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A function such as f(x) = x^2 has no inverse over all real numbers, because 3 and -3 give the same output.\"}}]}<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Function formulas explained simply: linear, quadratic, composite and inverse, with worked Indian examples using GST, taxi fares and factory profit.<\/p>\n","protected":false},"author":1,"featured_media":2915,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"footnotes":""},"categories":[24],"tags":[],"class_list":["post-2815","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Function Formula Explained with Examples<\/title>\n<meta name=\"description\" content=\"Function formulas explained simply: linear, quadratic, composite and 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