For each of the following functions find the domain and range:
A) f(x) = surd x
B) f(x) = 1/x^2
C) f(x) = surd 4-x
The hint is to sketch a graph for each function but how would you draw them with no coordinates given in the question??
A) f(x) = surd x
B) f(x) = 1/x^2
C) f(x) = surd 4-x
The hint is to sketch a graph for each function but how would you draw them with no coordinates given in the question??
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"Surd"! Don't use that word in the States much...
Assuming "surd" to mean square root (it can be others but let's go with that), let's explain why the answers are as such.
Domain means the set of all possible inputs for a function f(x) and range is the set of all possible outputs. Since you cannot take the square root (er, surd) of a negative number in the reals, its input must be nonnegative, so x >= 0. The output of a surd is the principal root--the positive real--so the range of f(x) is nonnegative and tends to infinity as x tends to infinity.
For the second one, the denominator equals 0 when x = 0, and you can't divide by 0, so 0 is excluded from the domain. Since x^2 is positive for all x in the reals, the fraction 1/x^2 is always positive, so the range is all y greater than 0. (At infinity, y = 0 theoretically, but let's go with what we can see.)
For the third one, it's the same argument as the first one; the input cannot be negative, so 4 - x >= 0 and therefore x <= 4. The range is the same as in the first one.
As for how to graph: the first function is the inverse of the right branch (x >= 0) of y = x^2, so if you sketch that lightly and then reflect it over the line y = x you' have the curve y = surd x. The third one is the same curve just shifted four units right.
For the second one, plot points at (1,1) and (2,1/4) as well as (-1,1) and (-2,1/4). Now the curve will approach (but not cross) the x-axis as you go out toward infinity and negative infinity, and the graph will shoot straight up near the y-axis as you approach 0 from either side; while the function is undefined there, think of it as equaling infinity.
Assuming "surd" to mean square root (it can be others but let's go with that), let's explain why the answers are as such.
Domain means the set of all possible inputs for a function f(x) and range is the set of all possible outputs. Since you cannot take the square root (er, surd) of a negative number in the reals, its input must be nonnegative, so x >= 0. The output of a surd is the principal root--the positive real--so the range of f(x) is nonnegative and tends to infinity as x tends to infinity.
For the second one, the denominator equals 0 when x = 0, and you can't divide by 0, so 0 is excluded from the domain. Since x^2 is positive for all x in the reals, the fraction 1/x^2 is always positive, so the range is all y greater than 0. (At infinity, y = 0 theoretically, but let's go with what we can see.)
For the third one, it's the same argument as the first one; the input cannot be negative, so 4 - x >= 0 and therefore x <= 4. The range is the same as in the first one.
As for how to graph: the first function is the inverse of the right branch (x >= 0) of y = x^2, so if you sketch that lightly and then reflect it over the line y = x you' have the curve y = surd x. The third one is the same curve just shifted four units right.
For the second one, plot points at (1,1) and (2,1/4) as well as (-1,1) and (-2,1/4). Now the curve will approach (but not cross) the x-axis as you go out toward infinity and negative infinity, and the graph will shoot straight up near the y-axis as you approach 0 from either side; while the function is undefined there, think of it as equaling infinity.