Functions：练习题

Given the functions \(f(x) = \frac{x+2}{3}\) and \(g(x) = x^2 - 5\), find the value of \(gf(4)\).

A function \(h\) is defined by \(h(x) = \frac{1}{x+2} + 5\) for \(x \in \mathbb{R}, x \ne -2\). State the range of \(h\).

The point \(P(a, b)\) lies on the graph of \(y = f(x)\). The graph is transformed to \(y = 5 - 2f(3x + 6)\). The corresponding point on the new graph is \((-1, 7)\). Find the values of \(a\) and \(b\).

Find the inverse function, \(f^{-1}(x)\), for the function defined by \(f(x) = 5 - 2x\) for \(x \in \mathbb{R}\).

The function \(f\) is defined by \(f(x) = 3x^2 - 12x + 7\) for the domain \(0 \le x \le 3\). State the range of \(f\).

The function f is defined by f(x) = \( \frac{2x - 5}{3} \) for x \( \in \mathbb{R} \). Find an expression for f⁻¹(x).

The graph of \(y = f(x)\) is transformed by a stretch with scale factor 2 parallel to the \(x\)-axis, followed by a translation of \(3\) units in the negative \(y\)-direction. Write the equation of the transformed graph in terms of \(f(x)\).

The graph of \(y = \sqrt{x}\) is transformed by a stretch with scale factor \(0.5\) parallel to the \(x\)-axis, followed by a reflection in the \(y\)-axis, and then a translation by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\). Write the equation of the transformed graph.

Functions \( f \) and \( g \) are defined for \( x \in \mathbb{R} \) by \( f(x) = 4x - 1 \) and \( g(x) = \frac{6}{x+2} \) for \( x \neq -2 \).

(a) Find the value of \( fg(1) \).
(b) Find an expression for \( f^{-1}(x) \).
(c) Find an expression for \( ff(x) \), giving your answer in its simplest form.
(d) Solve the equation \( g(x) = f(0) \).

Functions f and g are defined by f(x) = 3x+2 for x
∈
ℜ
and g(x) = x² - 4 for x
∈
ℜ
,
x ≥ 0.

(a) Find
fg(x).

(b) State the range of
fg.

(c) Solve the equation
fg(x) = gf(x).