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Sagot :
To find the domain for this function, we can see that the restriction we need to take into account is that the values in the radical must be values equal or greater than zero, so this function can have values in the Real set of numbers. Then, we have:
[tex]y=5\sqrt[]{2x-7}+10[/tex]We need to evaluate:
[tex]2x-7\ge0[/tex]Then, add 7 to both sides of the inequality, and then dividing the inequality by 2 (at both sides again) we have:
[tex]2x-7+7\ge0+7\Rightarrow2x+0\ge7\Rightarrow2x\ge7\Rightarrow\frac{2}{2}x\ge\frac{7}{2}_{}\Rightarrow x\ge\frac{7}{2}[/tex]We have that the values for the domain of this function are those for which are equal or greater than 7/2.
We can write the domain of this function in interval notation as follows:
[tex]D=\lbrack\frac{7}{2},\infty)[/tex]The important fact here is that for this function to have a domain and a range in the Real set, we need to have this restriction for this function.
The values of 5 and 10 are 'displacements' of a parent function and do not affect the values for this function to be in the Real Set of numbers.
For example, the value of 5 multiply the function, and the values for the range are greater (for x values) if the function was not multiplied by 5 ( and this does not affect, however, the values for the domain).
The value of 10 makes the function to be shifted 10 units above in the y-axis (and it does not affect the most important restriction found above). However, it does affect the values for the range in the function.
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