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Therefore, if $a<0$ , then the range is $(-\infty ,q)$ , meaning that $f\left(x\right)$ can be any real number less than $q$ . Equivalently, one could write that the range is $\{y\in \mathbb{R}:y<q\}$ .
For example, the domain of $g\left(x\right)=3\xb7{2}^{x+1}+2$ is $\{x:x\in \mathbb{R}\}$ . For the range,
Therefore the range is $\left\{g\right(x):g(x)\in [2,\infty \left)\right\}$ .
For functions of the form, $y=a{b}^{(x+p)}+q$ , the intercepts with the $x$ - and $y$ -axis are calculated by setting $x=0$ for the $y$ -intercept and by setting $y=0$ for the $x$ -intercept.
The $y$ -intercept is calculated as follows:
For example, the $y$ -intercept of $g\left(x\right)=3\xb7{2}^{x+1}+2$ is given by setting $x=0$ to get:
The $x$ -intercepts are calculated by setting $y=0$ as follows:
Since $b>0$ (this is a requirement in the original definition) and a positive number raised to any power is always positive, the last equation above only has a real solution if either $a<0$ or $q<0$ (but not both). Additionally, $a$ must not be zero for the division to be valid. If these conditions are not satisfied, the graph of the function of the form $y=a{b}^{(x+p)}+q$ does not have any $x$ -intercepts.
For example, the $x$ -intercept of $g\left(x\right)=3\xb7{2}^{x+1}+2$ is given by setting $y=0$ to get:
which has no real solution. Therefore, the graph of $g\left(x\right)=3\xb7{2}^{x+1}+2$ does not have a $x$ -intercept. You will notice that calculating $g\left(x\right)$ for any value of $x$ will always give a positive number, meaning that $y$ will never be zero and so the graph will never intersect the $x$ -axis.
Functions of the form $y=a{b}^{(x+p)}+q$ always have exactly one horizontal asymptote.
When examining the range of these functions, we saw that we always have either $y<q$ or $y>q$ for all input values of $x$ . Therefore the line $y=q$ is an asymptote.
For example, we saw earlier that the range of $g\left(x\right)=3\xb7{2}^{x+1}+2$ is $(2,\infty )$ because $g\left(x\right)$ is always greater than 2. However, the value of $g\left(x\right)$ can get extremely close to 2, even though it never reaches it. For example, if you calculate $g(-20)$ , the value is approximately 2.000006. Using larger negative values of $x$ will make $g\left(x\right)$ even closer to 2: the value of $g(-100)$ is so close to 2 that the calculator is not precise enough to know the difference, and will (incorrectly) show you that it is equal to exactly 2.
From this we deduce that the line $y=2$ is an asymptote.
In order to sketch graphs of functions of the form, $f\left(x\right)=a{b}^{(x+p)}+q$ , we need to determine four characteristics:
For example, sketch the graph of $g\left(x\right)=3\xb7{2}^{x+1}+2$ . Mark the intercepts.
We have determined the domain to be $\{x:x\in \mathbb{R}\}$ and the range to be $\left\{g\right(x):g(x)\in (2,\infty \left)\right\}$ .
The $y$ -intercept is ${y}_{int}=8$ and there is no $x$ -intercept.
$x$ | A | B | C |
-2 | 7,25 | 6,25 | 2,5 |
-1 | 3,5 | 2,5 | 1 |
0 | 2 | 1 | 0,4 |
1 | 1,4 | 0,4 | 0,16 |
2 | 1,16 | 0,16 | 0,064 |
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