函數、定義域、值域、複合與反函數：练习题

Find the \( y \)-intercept of the function \( f(x) = 3^{x+1} - 5 \).

Consider the function \( f(x) = \frac{x+2}{x-1} \) for \( x \ne 1 \). Calculate the value of the triple composite function \( (f \circ f \circ f)(3) \).

Consider the function \( f(x) = \frac{2x+3}{x-2} \), for \( x \neq 2 \). The graph of \( f \) is translated by the vector \( \begin{pmatrix} p \\ q \end{pmatrix} \) to produce the graph of the function \( g(x) = \frac{4x-13}{x-5} \), for \( x \neq 5 \). Find the value of \( p + q \).

The graph of a function \( y = f(x) \) is reflected in the \( y \)-axis. Which of the following equations represents the resulting graph?

The graph of \( f(x) = x^2 \) is stretched horizontally by a scale factor of \( 2 \), and then translated by the vector \( \begin{pmatrix} 0 \\ -3 \end{pmatrix} \). What is the equation of the resulting graph?

Find the coordinates of the y-intercept of the graph of the function \( f(x) = 3(x - 2)^2 + 7 \).

The quadratic function \( f(x) = 2x^2 - kx + 8 \) has no real zeros. Determine the range of possible values for the constant \( k \).

Consider the function $$f(x) = \frac{e^x - 1}{e^x + 1}$$. Determine the inverse function, $$f^{-1}(x)$$, and state its domain.

Given the functions \( f(x) = 2x - 3 \) and \( g(x) = x^2 + 1 \).

a) Find the expression for \( (f \circ g)(x) \).

b) Evaluate \( (g \circ f)(2) \).

Let \( f(x) = 2x^2 - 12x + 13 \).
a) Express \( f(x) \) in the form \( a(x-h)^2 + k \). (2 marks)
b) The graph of a function g is obtained by reflecting the graph of f in the y-axis, followed by a translation of the vector \( \begin{pmatrix} 2 \\ -1 \end{pmatrix} \). Find the coordinates of the vertex of the graph of g. (2 marks)<\/p>