∘

Function Operations

Add, subtract, multiply, divide — and compose.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualAdd, subtract, multiply, or divide two functions point-by-point. The new graph is the pointwise combination.

Algebra $(f+g)(x) = f(x)+g(x)$ , $(f-g)(x)=f(x)-g(x)$ , $(fg)(x)=f(x)g(x)$ , $\big(\tfrac{f}{g}\big)(x)=\tfrac{f(x)}{g(x)}$ where $g(x)\neq 0$ .

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

VisualFeed the output of one function into the input of another — a function inside a function.

Algebra $(f \circ g)(x) = f(g(x))$ . Apply $g$ first, then $f$ .

(f \circ g)(x) = f\big(g(x)\big)

VisualSpot the inner function — the thing you'd compute first if you plugged in a number.

AlgebraIf $h(x) = \sqrt{x^2+1}$ , then $h = f \circ g$ with $g(x)=x^2+1$ and $f(u)=\sqrt{u}$ .

VisualYou can only feed f values that g actually produces — and that f can accept.

Algebra $\text{dom}(f\circ g) = \{x \in \text{dom}(g) : g(x) \in \text{dom}(f)\}$ .

Sum

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

Add outputs

Composition

(f\circ g)(x) = f(g(x))

g first, then f

Domain of f∘g

\{x \in \text{dom}(g) : g(x) \in \text{dom}(f)\}

Restriction cascade

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easyoperations

f(x)=x+1

and

g(x)=x^2

, find

(f+g)(x)

easyoperations

f(x)=2x

and

g(x)=x+3

, find

f(g(1))

mediumoperations

f(x)=x^2

and

g(x)=x+1

, find

(f \circ g)(x)

mediumoperations

f(x)=\sqrt{x}

and

g(x)=x-4

, find the domain of

(f \circ g)(x)

hardoperations

Decompose

h(x)=\sqrt[3]{2x+5}

f \circ g

with

f(x)=\sqrt[3]{x}

. Find

g(x)