Functions, graphs and calculus：練習問題

What is the gradient of the straight line with the equation \( 4y - 12x = 8 \)?

The line \( L \) passes through the points \( (1, 4) \) and \( (3, 10) \).
Find the equation of the line that is parallel to \( L \) and passes through the origin \( (0, 0) \).

Find the gradient of the tangent to the curve \( y = 3x^2 - 4x + 2 \) at the point where \( x = 2 \).

The function f is defined by the expression:
\( f(x) = 4 - 3x \)

Find the value of \( x \) such that \( f(x) = 19 \).

The functions \( f \) and \( g \) are defined by \( f(x) = 3x + 2 \) and \( g(x) = x^2 - 4 \).
Find an expression for the composite function \( gf(x) \).

Given the function \( f(x) = \frac{x^2 - 4}{2} \), calculate the value of \( f(6) \).

Find the gradient of the curve \( y = 10x - x^2 \) at the point where \( x = 3 \).

A curve has the equation \( y = \frac{4}{x} + 2x \) for \( x \neq 0 \).
Find the coordinates of the two points on the curve where the gradient of the tangent is equal to 1.

The function f is defined by the expression:
\( f(x) = 15 - 4x \)

a) Calculate the value of \( f(2.5) \).
b) Solve the equation \( f(x) = -1 \).
c) Given that \( f(k) = k \), find the value of \( k \).

Two functions are defined as \( f(x) = \frac{x + 3}{x} \) for \( x \neq 0 \) and \( g(x) = 2x - 5 \).
a) Calculate the value of the composite function \( fg(4) \).
b) Find the values of \( x \) for which \( f(x) = g(x) \). Give your answers in the form \( \frac{p \pm \sqrt{q}}{r} \) where \( p, q, \) and \( r \) are integers.