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102 LearnersLast updated on September 26, 2025
Explicit Function
An explicit function shows a clear relationship between dependent and independent variables. It expresses the output variable directly in terms of the input variable. In this article, we'll learn about explicit functions and how they are used across various fields like physics, engineering, and physics.
What is an Explicit Function?
An explicit function is a mathematical expression where the dependent variable is clearly defined in terms of the independent variable. It’s commonly written as y = f(x), meaning the value of y depends directly on x. This form makes it easy to calculate the output for any given input.
Difference Between Implicit and Explicit Functions
Implicit and explicit functions differ in how the dependent variable is expressed in terms of the independent variable. In some cases, simplifying an implicit function can convert it into an explicit form. Here are a few differences between implicit and explicit functions.
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Implicit Functions |
Explicit Functions |
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In an implicit function, the relationship between variables is combined in one equation; the dependent variable is not isolated. |
In an explicit function, the dependent variable is expressed clearly in terms of the independent variable. |
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Represented as: f(x,y) = 0 |
Represented as: y = f(x) |
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Solving an implicit function requires rearranging or algebraic manipulation to isolate one variable. |
Directly substituting values into the equation gives the output. |
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The relationship between variables is unclear. |
The relationship between variables is straightforward. |
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For example, x2 + y2 = 25 (circle equation) |
For example: y = 3x2 - 2x + 1 (Quadratic function) |
Derivative of Explicit Function
The derivatives of explicit functions are found using basic rules of differentiation. Explicit functions are written as y = f(x), where x and y are the input and output, respectively, and the derivative of the function is y’ = f’(x). Let us take a few examples to understand how to find derivatives of an explicit function.
Example 1: Find the derivative of explicit function y = 3x3 - 5x + cos(x)
To differentiate y = 3x3 - 5x + cos(x), we apply standard differentiation rules with respect to x;
Hence, the derivative is dydx=9x2-5-sin(x)
Example 2: Find the derivative of x2y - 2y = 0
This is an implicit function, so we’ll use implicit differentiation.
Differentiate both sides with respect to x:
ddx (x2y - 2y) = ddx (0)
Applying the product rule on x2y, we get:
ddx (x2y) = x2 dydx + 2x y
ddx (-2y) = -2 dydx
Putting it all together, we get:
x2 dydx + 2xy - 2 dydx = 0
Now, combine the dydx terms:
(x2 - 2) dydx = -2xy
Now solve for dydx:
dydx = -2xyx2 - 2
Example 3: Find the derivative of the explicit function y = ln(x2 + 1)
Differentiate both sides with respect to x:
dydx=1x2+12x=2xx2+ 1
So, dydx=2xx2+ 1
Real-Life Applications of Explicit Function
Explicit functions help in making predictions and solving problems across various fields in real life. Some of these real-life applications are listed below.
Household budgeting
Explicit functions help in calculating expenses. For instance, the total cost (y) can be represented as the sum of a fixed rent of $7000 and $3000 for each unit (x):
y = 7000 + 3000x.
Motion calculations in physics
Distance is modeled as d = rt in uniform motion, where d is an explicit function of time t with constant rate r.
Profit and revenue analysis in a business
The equation for profit is P(x) = R(x) - C(x). If revenue and cost are both explicit, then P(x) becomes an explicit function.
Temperature conversion
Celsius is converted into Fahrenheit using the explicit function F = 95C+32.
Estimating material requirements in construction
Functions like y = 4x + 11 can be used to estimate the total amount of a material (y) needed to build a platform of length x meters, where 4 is the amount needed per meter and 11 is a fixed amount used at the start.
Common Mistakes and How to Avoid Them in Explicit Function
Although explicit functions are easier to work with than implicit functions, it can sometimes get difficult, especially when working with expressions involving trigonometric, logarithmic, or rational components. In these cases, students might make some mistakes. Knowing about these mistakes can help us avoid them in the future.
Mistake 1
Unable to differentiate whether a function is explicit or implicit.
It is common for students to get confused between implicit and explicit functions. However, this confusion can be avoided by remembering that in explicit functions, the dependent variable is written in terms of the independent variable, such as y = 3x + 9. On the other hand, implicit functions mix the variables, for instance 2x - y2 = 4.
Mistake 2
Assuming all functions are explicit by nature
Students forget that there are different types of functions, and not all functions are naturally explicit. Some require algebraic rearrangement to express the dependent variable clearly in terms of the independent variable. For instance, 2x2 + y = 1 needs to be solved for y to make it explicit.
Mistake 3
Finding derivatives incorrectly
Students often misapply rules of differentiation in explicit functions, which leads to incorrect results when calculating derivatives. Finding derivatives for explicit functions requires careful application of rules like power rule, product rule, quotient rule, and chain rule.
Mistake 4
Assuming linearity and graphing without checking behavior
One should always check for asymptotes, discontinuities, or turning points based on the form of the equation before graphing.
Mistake 5
Misinterpreting outcomes
Calculation errors are the most common among students, but can be easily avoided by following the correct order of operations(PEMDAS or BODMAS).
Solved Examples of Explicit Function
Problem 1
In y = 3x + 2, x = 4. Find the value of y.
y = 14
Explanation
Substitute x = 4 into the function:
y = 3(4) + 2
y = 14
Problem 2
The revenue of a company is given by R(x) = 50x. x is the quantity of sold items. Find the revenue for 20 items.
R(20) = $1000
Explanation
The function tells us that each item sold brings in $50. So multiplying 50 by the number of items (20) gives the total revenue. So:
R(20) = 50 × 20 = 1000.
Problem 3
Convert 27°C to Fahrenheit.
80.6°F
Explanation
This is an explicit formula for converting Celsius to Fahrenheit. We just substitute 27 for C, then follow the order of operations.
F = 95C + 32
F = 95(27)+32=48.6+32=80.6
Problem 4
If a cab charges a base fare of $4 and $3 per mile, write an explicit function for the same. Also find the fare for 10 miles.
Function f(x) = 3x + 4 and the fare will be $34
Explanation
The total cost is made of two parts: $3 for every mile (that’s 3x) and a one-time base charge of $4. Plugging in 10 miles gives the total fare.
For 10 miles, x = 10, F(x) = 3(10) + 4 = 30 + 4 = 34.
Problem 5
The height of a falling object is given by h(t) = 100 - 5t2. Find the height at t = 3 seconds.
h(3) = 55 meters
Explanation
This is an explicit function that models how an object falls over time. The term 5t2 shows that the height decreases more quickly as time increases. Substituting t = 3 gives the height after 3 seconds.
h(3) = 100 - 5(3)2 = 100 - 45 = 55.
FAQs on Explicit Function
1.Can all functions be written explicitly?
2.Which is the explicit formula?
3.What is meant by implicit?
4.What is an explicit function also called?
5. What is the explicit function in Excel?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
