AP®︎ Precalculus: Topic 2.7 Practice Test

Essential Knowledge: 2.7.A.1

If $f$ and $g$ are functions, the composite function $f \circ g$ maps a set of input values to a set of output values such that the output values of $g$ are used as input values of $f$. For this reason, the domain of the composite function is restricted to those input values of $g$ for which the corresponding output value is in the domain of $f$. $(f \circ g)(x)$ can also be represented as $f(g(x))$.

Essential Knowledge: 2.7.A.2

Values for the composite function $f \circ g$ can be calculated or estimated from the graphical, numerical, analytical, or verbal representations of $f$ and $g$ by using output values from $g$ as input values for $f$.

Essential Knowledge: 2.7.A.3

The composition of functions is not commutative; that is, $f \circ g$ and $g \circ f$ are typically different functions; therefore, $f(g(x))$ and $g(f(x))$ are typically different values.

Essential Knowledge: 2.7.A.4

If the function $f(x)=x$ is composed with any function $g$, the resulting composite function is the same as $g$; that is, $g(f(x)) = f(g(x)) = g(x)$. The function $f(x)=x$ is called the identity function. When composing two functions, the identity function acts in the same way as 0, the additive identity, when adding two numbers and 1, the multiplicative identity, when multiplying two numbers.

Essential Knowledge: 2.7.B.1

Function composition is useful for relating two quantities that are not directly related by an existing formula.

Essential Knowledge: 2.7.B.2

When analytic representations of the functions $f$ and $g$ are available, an analytic representation of $f(g(x))$ can be constructed by substituting $g(x)$ for every instance of $x$ in $f$.

Essential Knowledge: 2.7.B.3

A numerical or graphical representation of $f \circ g$ can often be constructed by calculating or estimating values for $(x, f(g(x)))$.

Essential Knowledge: 2.7.C.1

Functions given analytically can often be decomposed into less complicated functions. When properly decomposed, the variable in one function should replace each instance of the function with which it was composed.

Essential Knowledge: 2.7.C.2

An additive transformation of a function, $f$, that results in vertical and horizontal translations can be understood as the composition of $g(x)=x+k$ with $f$.

Essential Knowledge: 2.7.C.3

A multiplicative transformation of a function, $f$, that results in vertical and horizontal dilations can be understood as the composition of $g(x)=kx$ with $f$.