That means \(y\) represents the numbers
\[ \left [ \answer {-7}, \answer {11} \right ) \]
no pictures
The equation relating two functions tells us the algebraic transformations for the
domain and range.
We just need to read it correctly.
\(\blacktriangleright \) Suppose we have a function \(H\) with the set \([-5, 1) \cup (7, 11]\) as its domain and the set \((-3, 9]\) as its range.
Let’s use \(d\) to represent domain values of \(H\). That means \(d\) represents the numbers \([-5, 1) \cup (7, 11]\).
Then, the symbol \(H(d)\) represents the numbers \((-3, 9]\),
\(\blacktriangleright \) Now let’s introduce a second function called \(P\), which is related to \(H\).
Let’s choose \(k\) to represent the domain values of \(P\).
\(\blacktriangleright \) Let’s suppose that \(P\) and \(H\) are related by the equation
\[ P(k) = 4 H(k-1) - 2 \]
What can we say about \(P\)?
Domain
We would like to collect information about the domain of \(P\).
We know that \(P(k) = 4 H(k-1) - 2\).
\(k\) is representing the domain values of \(P\). If we can figure out what values \(k\) can have,
then we would know what values are in the domain of \(P\).
So, what do we know about \(k\)?
We know that \(k-1\) represents domain values of \(H\). We know this because the equation uses
the notation \(H(k-1)\). The expression inside the parentheses represents the domain values of \(H\).
That’s what function notation means.
That means \(k-1\) represents the values \([-5, 1) \cup (7, 11]\).
\[ k - 1 \in [-5, 1) \cup (7, 11] \]
\[ k \in [-4, 2) \cup (8, 12] \]
We now know the values that \(k\) can represent, which means we know the domain of \(P\).
Range
We would like to collect information about the range of \(P\).
In other words, we would like to know the function values of \(P\).
We know that \(P(k) = 4 H(k-1) - 2\).
Or, we know that \(P = 4 H - 2\).
We know that \(H\) represents the values \((-3, 9]\).
Then, \(4 H\) represents the values \((4 \cdot -3, 4 \cdot 9] = (-12, 36]\)
Then, \(4 H - 2\) represents the values \((-12 - 2, 36 - 2] = (-14, 34]\)
But \(P = 4 H - 2\). Therefore, \(P\) represents the values \((-14, 34]\).
The range of \(P\) is \((-14, 34]\).
\(\blacktriangleright \) Suppose we have a function \(m\) with the set \([-7, 11)\) as its domain and the set \((-12, 8]\) as its range.
Let’s use \(y\) to represent domain values of \(m\).
\(\blacktriangleright \) Now let’s introduce a second function called \(W\), which is related to \(m\).
Let’s choose \(t\) to represent the domain values of \(W\).
\(\blacktriangleright \) Let’s suppose that \(W\) and \(m\) are related by the equation
\[ W(t) = -2 m(3t + 2) - 5 \]
In this function equation, what is representing the domain values of \(W\)?
\(t\) \(k\) \(3k+2\) \(3t+2\)
\(\blacktriangleright \) Suppose we have a function \(m\) with the set \([-7, 11)\) as its domain and the set \((-12, 8]\) as its range.
Let’s use \(y\) to represent domain values of \(m\).
\(\blacktriangleright \) Now let’s introduce a second function called \(W\), which is related to \(m\).
Let’s choose \(t\) to represent the domain values of \(W\).
\(\blacktriangleright \) Let’s suppose that \(W\) and \(m\) are related by the equation
\[ W(t) = -2 m(3t + 2) - 5 \]
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more examples can be found by following this link
More Examples of Stretching