Composition of functions
A shop first applies a $15\%$ discount and then subtracts a fixed coupon of $20\,€$: the final price depends on the original price through two functions applied one after the other. This operation between functions, new in Bachillerato, is called composition, and the order in which the functions are applied changes the result.
What it means to compose two functions
Given the functions $f(x) = 3x + 2$ and $g(x) = x^2 - 1$, calculate the numerical value of the composite function $(g \circ f)(2)$.
Given the functions $f(x) = 3x + 2$ and $g(x) = x^2 - 1$, calculate the numerical value of the composite function $(g \circ f)(2)$.
Observe the following graph where the functions $f(x)$ (in blue) and $g(x)$ (in green) are represented. From the visual information and without needing to obtain their…
Observe the following graph where the functions $f(x)$ (in blue) and $g(x)$ (in green) are represented. From the visual information and without needing to obtain their algebraic expressions, determine the numerical values of $(f \circ g)(2)$ and $(g \circ f)(-1)$.
Algebraic expression of the composition
Find the algebraic expression for the composite function $(f \circ g)(x)$ given that $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$.
Find the algebraic expression for the composite function $(f \circ g)(x)$ given that $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$.
The daily production cost of $x$ parts in a factory is given by the function $C(x) = 15x + 500$ euros. The number of parts assembled by a machine in $t$ hours of…
The daily production cost of $x$ parts in a factory is given by the function $C(x) = 15x + 500$ euros. The number of parts assembled by a machine in $t$ hours of continuous operation is $x(t) = 40t$. Express the production cost as a function of the machine's operating time and calculate the total cost after a shift of $8$ hours.
Composition is not commutative
The shop context confirms it: discounting $15\%$ first and then subtracting $20\,€$ does not give the same final price as subtracting the $20\,€$ first and then applying the $15\%$.
Using the functions $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$, find the expression for $(g \circ f)(x)$ and prove analytically that function composition is not commutative.
Using the functions $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$, find the expression for $(g \circ f)(x)$ and prove analytically that function composition is not commutative.
An electronics shop applies a discount of $15\%$ to all its products, which is modelled by the function $d(x) = 0,85x$. In addition, a customer has a loyalty voucher for…
An electronics shop applies a discount of $15\%$ to all its products, which is modelled by the function $d(x) = 0,85x$. In addition, a customer has a loyalty voucher for a direct discount of $20$ euros, modelled by the function $c(x) = x - 20$. Determine the expressions for $(c \circ d)(x)$ and $(d \circ c)(x)$. Which order of composition benefits the customer more if they want to buy a computer with an initial price of $600$ euros?
Domain of the composite function
Rigorously determine the domain of the composite function $(g \circ f)(x)$ given that $f(x) = x - 4$ and $g(x) = \sqrt{x}$.
Rigorously determine the domain of the composite function $(g \circ f)(x)$ given that $f(x) = x - 4$ and $g(x) = \sqrt{x}$.
Consider the functions $f(x) = \sqrt{x + 3}$ and $g(x) = \frac{1}{x^2 - 4}$. Perform a rigorous analytical study to determine the domain of definition of the composite…
Consider the functions $f(x) = \sqrt{x + 3}$ and $g(x) = \frac{1}{x^2 - 4}$. Perform a rigorous analytical study to determine the domain of definition of the composite function $(f \circ g)(x)$.
Back to the beginning: If $f(x) = 0{,}85x$ represents applying the $15\%$ discount and $g(x) = x - 20$ subtracting the coupon, the shop applies $(g \circ f)(x) = 0{,}85x - 20$; for an item costing $100\,€$ you would pay $65\,€$, whereas with the reverse order $(f \circ g)(100) = 0{,}85 \cdot (100 - 20) = 68\,€$.