Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.
Here are some exam-style questions on this statement:
- "\(f(x) = \frac{2x}{5} + 7\) and \(g(x) = 10x^2 - 15\) for all values of \(x\)." ... more
- "(a) A function is represented by the following function machine." ... more
- "If \(f(x)=5-4x\) and \(g(x)=4^{-x}\) then:" ... more
- "Here is a function machine that produces two outputs, A and B." ... more
- "The functions \(f\) and \(g\) are such that:" ... more
- "Part of the function of \(y=f(x)\) is shown in the following diagram." ... more
- "The functions \(f\) and \(g\) are defined for \(x \in \mathbb{R} \) by" ... more
- "The diagram shows the graph of \(y=f(x)\), for \(-3\le x \le 4\)." ... more
- "The Big Wheel at Fantasy Fun Fayre rotates clockwise at a constant speed completing 15 rotations every hour. The wheel has a diameter of 90 metres and the bottom of the wheel is 6 metres above the ground." ... more
- "The functions f and g are defined as follows:" ... more
- "The functions \( f \) and \( g \) are defined for \( x \in \mathbb{R} \) by \( f(x) = 3 + 5x - 2x^2 \) and \( g(x) = x + k \), where \( k \in \mathbb{R} \)." ... more
- "The Fun Wheel at the Meller Theme Park rotates at a constant speed." ... more
- "Let \(f(x)=\frac{3x}{x-q}\), where \(x \neq q\)." ... more
- "Part of the graph of \(f(x) = {\log _b}(x + 4)\) for \(x > - 4\) is shown below." ... more
- "The functions \(f\) and \(g\) are defined as:" ... more
- "A function \( g \) is defined by \( g(x) = \frac{3x - 10}{x^2 - 9} \), where \( x \in \mathbb{R} \), \( x \neq \pm3 \)." ... more
- "The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the following equation:" ... more
Here is an Advanced Starter on this statement:
Click on a topic below for suggested lesson Starters, resources and activities from Transum.
Furthermore
Composite functions involve combining two or more functions to create a new function. The identity function, denoted as \( f(x) = x \), is a simple function where the output is the same as the input. Finding the inverse function, \( f^{-1}(x) \), involves swapping the x and y values in the original function and solving for y, which represents the new function \( f^{-1}(x) \).
The key formula for a composite function where two functions \( f \) and \( g \) are combined is given by: $$ (f \circ g)(x) = f(g(x)) $$ To find the inverse function \( f^{-1}(x) \), the equation \( y = f(x) \) is rearranged to make \( x \) the subject: $$ x = f^{-1}(y) $$
Example:
Let \( f(x) = 2x + 3 \) and \( g(x) = x - 1 \). The composite function \( (f \circ g)(x) \) and the inverse function \( f^{-1}(x) \) are found as follows:
Composite function:
$$ (f \circ g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x + 1 $$
Inverse function of \( f \):
Given \( y = 2x + 3 \), swap \( x \) and \( y \) and solve for \( y \):
$$ x = 2y + 3 \Rightarrow y = \frac{x - 3}{2} $$
So, \( f^{-1}(x) = \dfrac{x - 3}{2} \).
Transum,
Saturday, August 17, 2019
"There is an Advanced Lesson Starter called Permutable Functions that is open ended but allows students to consolidate their understanding of composite functions."
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