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Sagot :
For this, I am assuming that -25/2 is not a part of the exponent(although in future questions you should make this clear by using parenthesis.
When I do problems like this, I come to CBAD. where the variables are y=A(Bx+C)+D
CBAD helps significantly in domain and range problems where there is a well known parent function. (y=x^2, y=sqrt(x), y=1/x, y=x, etc). Basically, if there is a parent function with slight additions(y=2*x^2, y=sqrt(x+1), y=1/(5x), y=3x, etc)
This works that in the order, a C term shifts the graph horizontally in the opposite direction expected (a negative number will shift right, a positive number will shift left). For example, if C is -2, the graph of the parent function will shift right 2.
Then, a B term dilates(changes the horizontal size of the graph) by the factor 1/B(every x value is multiplied by 1/B). Thus a B >1 will shrink the graph horizontally, and a 0<B<1 with stretch the graph horizontally. If B is negative, it reflects the graph over the y axis and then either shrinks or stretches horizontally. If B is 1, nothing changes.
Then, a A term dilates(changes the horizontal size of the graph) by the factor A(every y value is multiplied by A). Thus a A >1 will stretch the graph vertically, and a 0<A<1 with shrink the graph vertically. If A is negative, it reflects the graph over the x axis and then either shrinks or stretches vertically. If A is 1, nothing changes.
Finally, a D term shifts the graph vertically in the direction expected (a negative number will shift down, a positive number will shift up). For example, if D is -2, the graph of the parent function will shift down 2.
Warning CBAD must be performed in the order C-B-A-D
Now that you know these simple rules and can memorize the few parent functions, you can figure out the domain and range of almost any function.
In your case, the function x^(2)-25/2 is not much different than x^2, an upwards facing parabola centered at (0,0). In fact, there is only one transformation, D, because C=0 (it is not shown), B=1(it is not shown), and A=1(it is not shown). Thus the only difference between your graph and the parent function is that yours is shifted down by 25/2.
For this reason, because the domain of x^2 is not restricted (it includes all real numbers), the domain of your new function is not restricted (it includes all real numbers).
When I do problems like this, I come to CBAD. where the variables are y=A(Bx+C)+D
CBAD helps significantly in domain and range problems where there is a well known parent function. (y=x^2, y=sqrt(x), y=1/x, y=x, etc). Basically, if there is a parent function with slight additions(y=2*x^2, y=sqrt(x+1), y=1/(5x), y=3x, etc)
This works that in the order, a C term shifts the graph horizontally in the opposite direction expected (a negative number will shift right, a positive number will shift left). For example, if C is -2, the graph of the parent function will shift right 2.
Then, a B term dilates(changes the horizontal size of the graph) by the factor 1/B(every x value is multiplied by 1/B). Thus a B >1 will shrink the graph horizontally, and a 0<B<1 with stretch the graph horizontally. If B is negative, it reflects the graph over the y axis and then either shrinks or stretches horizontally. If B is 1, nothing changes.
Then, a A term dilates(changes the horizontal size of the graph) by the factor A(every y value is multiplied by A). Thus a A >1 will stretch the graph vertically, and a 0<A<1 with shrink the graph vertically. If A is negative, it reflects the graph over the x axis and then either shrinks or stretches vertically. If A is 1, nothing changes.
Finally, a D term shifts the graph vertically in the direction expected (a negative number will shift down, a positive number will shift up). For example, if D is -2, the graph of the parent function will shift down 2.
Warning CBAD must be performed in the order C-B-A-D
Now that you know these simple rules and can memorize the few parent functions, you can figure out the domain and range of almost any function.
In your case, the function x^(2)-25/2 is not much different than x^2, an upwards facing parabola centered at (0,0). In fact, there is only one transformation, D, because C=0 (it is not shown), B=1(it is not shown), and A=1(it is not shown). Thus the only difference between your graph and the parent function is that yours is shifted down by 25/2.
For this reason, because the domain of x^2 is not restricted (it includes all real numbers), the domain of your new function is not restricted (it includes all real numbers).
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