2.1 Basic functions (composite)

The table below shows some functions values of [math]f(x)[/math] and [math]g(x)[/math]

Evaluate:
(a) [math](f\circ g\circ f)(1)[/math]
(b) [math]g(f(3)-1)[/math]
(c) [math]f(f(f(2)))[/math]

The table below shows some values of functions [math]f(x)[/math] and [math]g(x)[/math]. If [math]f(g(m))=4[/math], what is the value of [math]m[/math]?

Let [math]f(x)=-2x+3[/math] be a function. Define [math]f^n(x)[/math] as a function of [math]f(x)[/math] self-compositing [math]n[/math] times where [math]n\in\mathbb{N}[/math]. (ex: [math]f^3(x)=(f\circ f\circ f)(x)[/math])

Let [math]f(x)[/math] be a function satisfying [math]f(x+3)=2f(x)+1[/math]. Suppose the [math]x[/math]-intercept of [math]f(x)[/math] is 2. Find [math]f(11)[/math].

The figure below shows the graph of function [math]f(x)[/math]. The domain of [math]f(x)[/math] is [math][-6,6][/math] and [math]f(-1.5)=2[/math].

(a) Find all possible values of [math]m[/math] such that [math]f(m)=3[/math]. Write your answers in ascending order.

(b) Find the conditions of [math]k[/math] such that the equation [math]f(x)=k[/math] has 4 solutions.

(c) Evaluate [math]f(f(5))[/math].

(d) If [math]g(x)[/math] is a linear function and [math]g(x)[/math] and [math]f(x)[/math] have common [math]x[/math]-intercept and [math]y[/math]-intercept, what is the equation of [math]g(x)[/math]?

(e) Find the domain of [math]f(g(x))[/math].

(f) Solve the inequality [math]f(x)<g(x)[/math].

Let [math]f(x)[/math] be a linear function satisfying [math]f(2a+3)=4a-7[/math]. What is the summation of the slope and the [math]y[/math]-intercept of [math]f(x)[/math]?

Let [math]f(x)=-2|x|+3[/math] and [math]g(x)=\sqrt{x}-5[/math].

(a) Evaluate [math]f(g(4))[/math].

(b) Find the [math]x[/math]-intercept(s) of [math]f(g(x))[/math]. If you have two or more answers, write the smallest one.

If a linear function [math]f[/math] has the property [math]f(2x+1)=2x-1[/math], what is the slope of [math]f[/math]?

If [math]f(x)=px+q[/math] and [math]f(f(f(x)))=8x+21[/math], determine [math]p[/math] and [math]q[/math] where [math]p,q\in\mathbb{R}[/math].

If [math]f[/math] is a function such that [math]f(1)=2[/math] and [math]f(n+1)=\frac{3f(n)+1}{3}[/math] for [math]n=1,2,3,…[/math], what is the value of [math]f(100)[/math]?

Suppose [math]f(x)[/math] is a real-valued function such that [math]5f(\frac{1}{x})+\frac{f(2x)}{x^2}=x[/math] for [math]x\neq0[/math]. Find [math]f(1)[/math]

If [math]f(x)=7x+9[/math] and [math]g(x)=4x-1[/math], what is [math](f\circ g)(x)[/math]?

If [math]f(x)=\displaystyle{\frac{6}{x-6}}[/math] and [math]g(x)=\displaystyle{\frac{5}{4x}}[/math], what is [math](g\circ f)(x)[/math]?

If [math]f(x)=\{(-5,2),(-2,5),(-1,-5),(0,3),(2,-1)\}[/math], then [math]f(f(2))=[/math]

If [math]f(x)=\sqrt{x+3}[/math] and [math]g(x)=2x[/math], then [math](f\circ g)(4)=[/math]

Given [math]g(x)=2x^3+3[/math] and [math]h(x)=e^{3x}-2[/math]

(a) Find [math]h(0)[/math]

(b) Find [math](g\circ h)(0)[/math]

If [math]f(x+1)=xf(x)[/math], and [math]f(1)=1[/math], then [math]f(5)=[/math]?