02.05 Functions and Introduction to Graphs
In this lesson you will learn:
- the meaning of the terms function, domain, range and roots
- how to define functions and become familiar with function notation
- to use substitutions to solve disguised / hidden quadratics functions
Prerequisites from previous chapters:
- Laws of Indices
- Expanding brackets
VIDEO LESSON
INTERACTIVE SELF-STUDY
A function is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output.
- The notation \(f(x)\) is used to represent a function of \(x\)
- If \(a\) is an input then \(f(a)\) is the output.
- To calculate \(f(a)\), we replace all instances of \(x\) with \(a\)
For example if \(f(x)=x^2+3x+2\), and we want to input \(6\), then the output is \(f(6)= (6)^2+3(6)+2=56\)
Or if we want to input \(5c\) then the output is \(f(5c)=(5c)^2+3(5c)+2=25c^2+15c+2\)
Example 1 – (GRADE E)
Given that \(f(x)=7x^2-11x+13\), find:
a. \(f(3)\)
You can use the function button on your calculator to help you quickly compute the output of a function when given a numerical input. (You cannot use the calculator for algebraic inputs). In this way you are much less likely to make computational errors
For Example 1a, to work out \(f(3)\) you would press the following:
\(3 \rightarrow\) STO \(\rightarrow x \rightarrow\) AC \(\rightarrow 7x^2-11x+13 \rightarrow\) =
To change the input value for the same function e.g. to now work out \(f(5)\), you would do the following:
AC \(\rightarrow 5 \rightarrow\) STO \(\rightarrow x \rightarrow\) UP ARROW (so that it displays \(7x^2-11x+13\)) \(\rightarrow\) =
Notice how the ON button is never used, and with good reason. When the ON button is pressed, it completely clears your calculation history, so you would not be able to use the UP ARROW to redisplay \(7x^2-11x+13\). You would have to type out the expression again from scratch.
NEVER, EVER press the ON button, always use the AC button instead!
Example 2 – (GRADE E)
Given that \(g(x)=4x^2-x+1\), use your calculator to evaluate
a. \(g(4)\)
- The domain of a function is the set of possible input values.
If the input of a function, \(x\), can be any real number the domain can be written as \(x∈ \mathbb{R}\) .
We read this as ‘\(x\) is an element/a member of the real numbers’
- The range of a function is the set of possible output values.
The diagram adjacent shows how the function \(f(x)=x^2\) maps some values from its domain to values in its range.
Notice how each input has only one output.
For example it shows us that \(f(2)=4\)
However it is possible for two different inputs to yield the same output e.g. the inputs \(1\) and \(−1\) both yield \(1\)
Example 3 – (GRADE D)
The function \(f\) is given by \(f(x)=2x^2-3x+4\), \(x∈ \mathbb{R}\)
a. Find \(f(7)\)
- The roots of a function are the solutions to (values of \(x\) that satisfy) the equation \(f(x)=0\).
Example 4 – (GRADE D)
The function \(f\) is defined by \(f(x)=x^2+3x+2\), \(x∈ \mathbb{R}\)
a. Find the roots of \(f(x)\)
DISGUISED QUADRATICS
Some functions may be viewed as hidden/disguised quadratics. This is a function that does not initially appear to be a quadratic equation but can be transformed into one. They are of the form \(ax^2+bx+c\), but \(x\) has been replaced by some other variable/term.
For example \(y^4-5y^2-36\) is a hidden quadratic.
Notice the relationship between \(y^2\) and \(y^4\) is that \(y^4=(y^2)^2\).
So it seems as though we started with the function \(x^2-5x-36\), but then replaced every \(x\) with \(y^2\)!
- To solve hidden quadratic equations, you can use a substitution to more clearly reveal the underlying quadratic equation.
For \(y^4-5y^2-36=0\), we can use the substitution \(u=y^2\) to reduce it to \(u^2-5u-36=0\), then solve for the possible values of \(u\):
Letting \(u=y^2 \Rightarrow u^2-5u-36=0 \Rightarrow (u+4)(u-9)=0 \Rightarrow u=-4, u=9\)
We must then remember \(u=y^2 \Rightarrow y^2=-4\), \(y^2=9\)
Then we inspect whether these values of \(y^2\) are actually possible!
Since \(y^2≥0\), \(y^2=-4\) is invalid but \(y^2=9\) is valid
Now we solve for \(y\): \(y^2=9 \Rightarrow y=\pm3\)
Example 5 – (GRADE C)
Find the roots of the function \(f(x)=x^6-26x^3-27\), \(x∈ \mathbb{R}\)
END OF LESSON!
You can now answer
EDEXCEL Pure Year 1 Ex 2E – All Questions