Find F+g Calculator

Enter the coefficients of your polynomial functions f(x) and g(x) (up to x³) and the value of x at which to evaluate their sum (f+g)(x).

Function f(x) = ax³ + bx² + cx + d

f(x) = 1x³ + 2x² + 3x + 4

Function g(x) = ex³ + fx² + gx + h

g(x) = 0x³ + 1x² – 1x + 2

Term	f(x) Coeff.	g(x) Coeff.	(f+g)(x) Coeff.
x^3	1	0	1
x^2	2	1	3
x	3	-1	2
Constant	4	2	6

Coefficients of f(x), g(x), and (f+g)(x)

What is a Sum of Functions Calculator (f+g)(x)?

A Sum of Functions Calculator (f+g)(x) is a tool used to find the sum of two functions, denoted as f(x) and g(x), at a specific point 'x' or to find the expression for the combined function (f+g)(x). In algebra, when you have two functions f(x) and g(x), their sum (f+g)(x) is simply defined as f(x) + g(x). This means you add the outputs of the two functions for the same input value x, or you add the expressions of the two functions together by combining like terms.

This calculator is particularly useful for students learning about the algebra of functions, engineers, scientists, and anyone who needs to combine the effects or outputs of two different processes or models represented by functions. Our Sum of Functions Calculator (f+g)(x) handles polynomial functions up to the third degree, allowing you to quickly see the result of adding them.

Common misconceptions include thinking that (f+g)(x) is the same as f(g(x)) (which is function composition) or f(x) * g(x) (which is the product of functions). The Sum of Functions Calculator (f+g)(x) specifically deals with addition.

Sum of Functions (f+g)(x) Formula and Mathematical Explanation

The formula for the sum of two functions f(x) and g(x) is:

(f+g)(x) = f(x) + g(x)

To find the sum function (f+g)(x), you simply add the expressions for f(x) and g(x) together and combine like terms. For example, if f(x) = 2x + 1 and g(x) = x² – 3, then:

(f+g)(x) = (2x + 1) + (x² – 3) = x² + 2x – 2

To evaluate the sum at a specific point x=a, you calculate f(a) and g(a) and add them: (f+g)(a) = f(a) + g(a).

Variables in the Sum of Functions

Variable	Meaning	Unit	Typical Range
f(x), g(x)	The two functions being added	Depends on the context of the functions	Any real-valued functions
(f+g)(x)	The sum function	Same as f(x) and g(x)	The resulting sum function
x	The independent variable or input value	Depends on the context	Real numbers within the domain of both f and g
Coefficients	Constants multiplying the powers of x in polynomials	Dimensionless (if x is)	Real numbers

Practical Examples (Real-World Use Cases)

Let's look at a couple of examples using the Sum of Functions Calculator (f+g)(x) concept.

Example 1: Combining Cost Functions

Suppose a company has two cost components for producing an item: material cost f(x) = 10x + 50 (where x is the number of items) and labor cost g(x) = 0.5x² + 5x + 100.

The total cost function (f+g)(x) is f(x) + g(x) = (10x + 50) + (0.5x² + 5x + 100) = 0.5x² + 15x + 150.

If they produce x=10 items, f(10) = 100+50=150, g(10)=50+50+100=200, so (f+g)(10) = 150+200=350. Using the sum function: 0.5(100)+15(10)+150 = 50+150+150=350.

Example 2: Overlapping Signals

Imagine two simple wave functions f(t) = sin(t) and g(t) = 0.5sin(2t) representing signals. The combined signal is (f+g)(t) = sin(t) + 0.5sin(2t). While our calculator uses polynomials, the principle is the same. At t=π/2, f(π/2)=1, g(π/2)=0, so (f+g)(π/2)=1.

Using our Sum of Functions Calculator (f+g)(x) with polynomials: Let f(x) = x² – x and g(x) = 3x + 2. Find (f+g)(3).

f(x) coefficients: a=0, b=1, c=-1, d=0. g(x) coefficients: e=0, f=0, g=3, h=2. x=3.

f(3) = 9-3=6, g(3) = 9+2=11. (f+g)(3) = 6+11=17.

(f+g)(x) = x² + 2x + 2. At x=3, (f+g)(3) = 9+6+2 = 17.

How to Use This Sum of Functions Calculator (f+g)(x)

1. Enter f(x) coefficients: Input the coefficients for the x³, x², x, and constant terms of your first function, f(x).
2. Enter g(x) coefficients: Do the same for your second function, g(x). The calculator will display the functions based on your inputs.
3. Enter x value: Input the value of x at which you want to evaluate the sum.
4. View Results: The calculator instantly shows f(x), g(x), and the primary result (f+g)(x) at the given x. It also displays the expression for (f+g)(x).
5. Examine Table and Chart: The table shows the coefficients of f(x), g(x), and (f+g)(x). The chart visualizes the three functions around the entered x-value.
6. Reset or Copy: Use the "Reset" button to go back to default values or "Copy Results" to copy the output.

Understanding the results helps in seeing how the two functions combine their values at a specific point or how their expressions merge.

Key Factors That Affect Sum of Functions (f+g)(x) Results

• Coefficients of f(x): The values of the coefficients in f(x) directly determine the shape and values of f(x), thus affecting the sum.
• Coefficients of g(x): Similarly, the coefficients of g(x) determine its contribution to the sum.
• The value of x: The specific point 'x' at which the functions are evaluated dictates the output values of f(x), g(x), and consequently (f+g)(x).
• Degree of the Polynomials: Higher-degree terms can dominate the function's behavior for large |x|, significantly impacting the sum.
• Like Terms: When adding f(x) and g(x), only like terms (terms with the same power of x) are combined, affecting the structure of (f+g)(x).
• Domain of f(x) and g(x): The domain of (f+g)(x) is the intersection of the domains of f(x) and g(x). If x is outside this intersection, the sum is undefined. For polynomials, the domain is all real numbers.

Our Sum of Functions Calculator (f+g)(x) assumes polynomial functions where the domain is all real numbers, simplifying domain considerations.

Frequently Asked Questions (FAQ)

What is (f+g)(x)?

It represents the sum of two functions, f(x) and g(x), defined as (f+g)(x) = f(x) + g(x). It's a way of combining functions through addition.

How do you find the sum of two functions?

To find the expression for (f+g)(x), you add the expressions of f(x) and g(x) and combine like terms. To find the value at a point, calculate f(x) and g(x) at that point and add the results.

What is the domain of (f+g)(x)?

The domain of (f+g)(x) is the intersection of the domains of f(x) and g(x). Both functions must be defined at x for their sum to be defined.

Is (f+g)(x) the same as f(x)g(x)?

No, (f+g)(x) is the sum f(x) + g(x), while f(x)g(x) or (fg)(x) is the product of the two functions.

Is (f+g)(x) the same as f(g(x))?

No, f(g(x)) is the composition of f with g, where the output of g(x) becomes the input for f.

Can I use this calculator for non-polynomial functions?

This specific Sum of Functions Calculator (f+g)(x) is designed for polynomial functions up to the third degree because you input coefficients. The concept of (f+g)(x) = f(x) + g(x) applies to all functions, though.

What if one function is not defined at x?

If either f(x) or g(x) is not defined at a certain value of x, then (f+g)(x) is also not defined at that value of x.

How does the graph of (f+g)(x) relate to f(x) and g(x)?

At any x-value, the y-value of the (f+g)(x) graph is the sum of the y-values of the f(x) and g(x) graphs. You can think of it as vertically adding the graphs.