Algebra and functions：練習問題

The function \( f \) is defined by \( f(x) = x^2 - 2x \) for the domain \( 0 \le x \le 3 \). The function \( g \) is defined by \( g(x) = |x| \) for all real values of \( x \). Find the range of the composite function \( g(f(x)) \).

Solve the simultaneous equations:
\( y = x + 1 \)
\( y = x^2 - 5x + 6 \)

The graph of \( y = f(x) \) is transformed to the graph of \( y = 2f(x - 3) \). Describe the sequence of transformations.

Find the value of the discriminant for the quadratic equation \( 2x^2 - 5x + 3 = 0 \).

Solve the simultaneous equations:
\(y = 2x + 3\)
\(x^2 + y^2 = 5\)
Which of the following is a possible value for \(x\)?

Simplify the expression \((3x^2)^3 \times (2x^{-1})^2\).

Given that \( (x-2) \) is a factor of the polynomial \( f(x) = 2x^3 + ax^2 - 5x - 6 \), find the value of the constant \( a \) and hence factorise \( f(x) \) completely.
[4 marks]

Given the functions \( f(x) = e^{x} - 2 \) and \( g(x) = \ln(x + 5) \) for \( x > -5 \), find the exact value of \( x \) such that \( fg(x) = 6 \).

Find the set of values of \(x\) for which \((2x-1)(x+3) > 5\).