Domain and range of a function exercises with answers

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Domain and range of a function exercises with answers

The domain of a function is the complete set of possible values of the independent variable. In plain English, this definition means: The domain is the set of all possible x-values which will make the function "work", and will output real y-values. When finding the domain, remember: The denominator (bottom) of a fraction cannot be zero The number under a square root sign must be positive in this section Example 1a Here is the graph of `y = sqrt(x+4)`: 1 2 3 4 5 -1 -2 -3 -4 1 2 3 x y Domain: `x>=-4` The domain of this function is `x -4`, since x cannot be less than ` -4`. To see why, try out some numbers less than `-4` (like ` -5` or ` -10`) and some more than `-4` (like ` -2` or `8`) in your calculator. The only ones that "work" and give us an answer are the ones greater than or equal to ` -4`. This will make the number under the square root positive. Notes: The enclosed (colored-in) circle on the point `(-4, 0)`. This indicates that the domain "starts" at this point. We saw how to draw similar graphs in section 4, Graph of a Function. For a more advanced discussion, see also How to draw y^2 = x - 2. How to find the domain In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign). Range The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain. In plain English, the definition means: The range is the resulting y-values we get after substituting all the possible x-values. How to find the range The range of a function is the spread of possible y-values (minimum y-value to maximum y-value) Substitute different x-values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?) Make sure you look for minimum and maximum values of y. Draw a sketch! In math, it's very true that a picture is worth a thousand words. Example 1b Let's return to the example above, `y = sqrt(x + 4)`. We notice the curve is either on or above the horizontal axis. No matter what value of x we try, we will always get a zero or positive value of y. We say the range in this case is y 0. 1 2 3 4 5 -1 -2 -3 -4 1 2 3 x y Range: `y>=0` The curve goes on forever vertically, beyond what is shown on the graph, so the range is all non-negative values of `y`. Example 2 The graph of the curve y = sin x shows the range to be betweeen -1 and 1. 1 2 3 4 5 -1 -2 -3 -4 -5 -6 -7 1 -1 x y Range: `-1 The domain of y = sin x is "all values of x", since there are no restrictions on the values for x. (Put any number into the "sin" function in your calculator. Any number should work, and will give you a final answer between -1 and 1.) From the calculator experiment, and from observing the curve, we can see the range is y betweeen -1 and 1. We could write this as -1 y 1. Where did this graph come from? We learn about sin and cos graphs later in Graphs of sin x and cos x Note 1: Because we are assuming that only real numbers are to be used for the x-values, numbers that lead to division by zero or to imaginary numbers (which arise from finding the square root of a negative number) are not included. The Complex Numbers chapter explains more about imaginary numbers, but we do not include such numbers in this chapter. Note 2: When doing square root examples, many people ask, "Don't we get 2 answers, one positive and one negative when we find a square root?" A square root has at most one value, not two. See this discussion: Square Root 16 - how many answers? Note 3: We are talking about the domain and range of functions, which have at most one y-value for each x-value, not relations (which can have more than one.). Finding domain and range without using a graph It's always a lot easier to work out the domain and range when reading it off the graph (but we must make sure we zoom in and out of the graph to make sure we see everything we need to see). However, we don't always have access to graphing software, and sketching a graph usually requires knowing about discontinuities and so on first anyway. As meantioned earlier, the key things to check for are: There are no negative values under a square root sign There are no zero values in the denominator (bottom) of a fraction Example 3 Find the domain and range of the function `f(x)=sqrt(x+2)/(x^2-9),` without using a graph. Solution In the numerator (top) of this fraction, we have a square root. To make sure the values under the square root are non-negative, we can only choose `x`-values grater than or equal to -2. The denominator (bottom) has `x^2-9`, which we recognise we can write as `(x+3)(x-3)`. So our values for `x` cannot include `-3` (from the first bracket) or `3` (from the second). We don't need to worry about the `-3` anyway, because we dcided in the first step that `x >= -2`. So the domain for this case is `x >= -2, x != 3`, which we can write as `[-2,3)uu(3,oo)`. To work out the range, we consider top and bottom of the fraction separately. Numerator: If `x=-2`, the top has value `sqrt(2+2)=sqrt(0)=0`. As `x` increases value from `-2`, the top will also increase (out to infinity in both cases). Denominator: We break this up into four portions: When `x=-2`, the bottom is `(-2)^2-9=4-9=-5`. We have `f(-2) = 0/(-5) = 0.` Between `x=-2` and `x=3`, `(x^2-9)` gets closer to `0`, so `f(x)` will go to `-oo` as it gets near `x=3`. For `x>3`, when `x` is just bigger than `3`, the value of the bottom is just over `0`, so `f(x)` will be a very large positive number. For very large `x`, the top is large, but the bottom will be much larger, so overall, the function value will be very small. So we can conclude the range is `(-oo,0]uu(oo,0)`. Have a look at the graph (which we draw anyway to check we are on the right track): Show graph We can see in the following graph that indeed, the domain is `[2,3)uu(3,oo)` (which includes `-2`, but not `3`), and the range is "all values of `f(x)` except `F(x)=0`." Graph of `f(x)=sqrt(x+2)/(x^2-9)`. Summary In general, we determine the domain by looking for those values of the independent variable (usually x) which we are allowed to use. (We have to avoid 0 on the bottom of a fraction, or negative values under the square root sign). The range is found by finding the resulting y-values after we have substituted in the possible x-values. Exercise 1 Find the domain and range for each of the following. (a) `f(x) = x^2+ 2`. Answer Domain: The function f(x) = x2 + 2 is defined for all real values of x (because there are no restrictions on the value of x). Hence, the domain of `f(x)` is "all real values of x". Range: Since x2 is never negative, x2 + 2 is never less than `2` Hence, the range of `f(x)` is "all real numbers `f(x) 2`". We can see that x can take any value in the graph, but the resulting y = f(x) values are greater than or equal to 2. 1 2 3 -1 -2 -3 1 2 3 4 5 6 7 8 9 10 -1 x f(x) Range: `y>=2` Domain: All `x` Note It is important to label the axes when sketching graphs. It helps with understanding what the graph represents. We saw how to sketch such graphs in Graph of a Function. Download graph paper (b) `f(t)=1/(t+2)` Answer Domain: The function `f(t)=1/(t+2)` is not defined for t = -2, as this value would result in division by zero. (There would be a 0 on the bottom of the fraction.) Hence the domain of f(t) is "all real numbers except -2" Range: No matter how large or small t becomes, f(t) will never be equal to zero. [Why? If we try to solve the equation for 0, this is what happens: `0=1/(t+2)` Multiply both sides by (t + 2) and we get `0 = 1` This is impossible.] So the range of f(t) is "all real numbers except zero". We can see in the graph that the function is not defined for `t = -2` and that the function (the y-values) takes all values except `0`. 1 2 3 4 -1 -2 -3 -4 -5 6 -7 1 2 3 4 5 -1 -2 -3 -4 -5 t f(t) Domain: All `t -2` Range: All `f(t) 0` (c) `g(s)=sqrt(3-s)` Answer The function `g(s)=sqrt(3-s)` is not defined for real numbers greater than 3, which would result in imaginary values for g(s). Hence, the domain for g(s) is "all real numbers, s 3". Also, by definition, `g(s)=sqrt(3-s) >= 0` Hence, the range of g(s) is "all real numbers `g(s) 0`" We can see in the graph that s takes no values greater than 3, and that the range is greater than or equal to `0`. 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -7 1 2 3 4 -1 -2 s g(s) Domain: All `s Range: All `g(s) >= 0` (d) `f(x) = x^2+ 4` for `x > 2` Answer The function `f(x)` has a domain of "all real numbers, `x > 2`" as defined in the question. (There are no resulting square roots of negative numbers or divisions by zero involved here.) To find the range: When `x = 2`, `f(2) = 8` When x increases from `2`, `f(x)` becomes larger than `8` (Try substituting in some numbers to see why.) Hence, the range is "all real numbers, `f(x) > 8`" Here is the graph of the function, with an open circle at `(2, 8)` indicating that the domain does not include `x = 2` and the range does not include `f(2) = 8`. 1 2 3 4 5 6 5 10 15 20 25 x f(x) (2, 8) Domain: All `x>2` Range: All `f(x) > 8` The function is part of a parabola. [See more on parabola.] Exercise 2 We fire a ball up in the air and find the height h, in metres, as a function of time t, in seconds, is given by h = 20t - 4.9t2 Find the domain and range for the function h(t). Answer Generally, negative values of time do not have any meaning. Also, we need to assume the projectile hits the ground and then stops - it does not go underground. So we need to calculate when it is going to hit the ground. This will be when h = 0. So we solve: 20t - 4.9t2 = 0 Factoring gives: (20 - 4.9t)t = 0 This is true when `t = 0\ "s"`, or `t=20/4.9 = 4.082 text(s)` Hence, the domain of the function h is "all real values of t such that `0 t 4.082`" We can see from the function expression that it is a parabola with its vertex facing up. (This makes sense if you think about throwing a ball upwards. It goes up to a certain height and then falls back down.) What is the maximum value of h? We use the formula for maximum (or minimum) of a quadratic function. The value of t that gives the maximum is `t = -b/(2a) = -20/(2 xx (-4.9)) = 2.041 s ` So the maximum value is 20(2.041) - 4.9(2.041)2 = 20.408 m By observing the function of h, we see that as t increases, h first increases to a maximum of 20.408 m, then h decreases again to zero, as expected. Hence, the range of h is "all real numbers, `0 h 20.408`" Here is the graph of the function h: 1 2 3 4 5 6 5 10 15 20 -5 t h(t) Domain: `0 Range:`0 Functions defined by coordinates Sometimes we don't have continuous functions. What do we do in this case? Let's look at an example. Exercise 3 Find the domain and range of the function defined by the coordinates: `{(-4, 1), (-2, 2.5), (2, -1), (3, 2)}` Answer The domain is simply the x-values given: `x = {-4, -2, 2, 3}` The range consists of the `f(x)`-values given: `f(x) = {-1, 1, 2, 2.5}` Here is the graph of our discontinuous function. 1 2 3 4 -1 -2 -3 -4 1 2 3 4 -1 -2 -3 t h(t) (3, 2) (2, -1) (-4, 1) (-2, 2.5) Page 2 By M. Bourne Even Functions A function `y = f(t)` is said to be even if f(-t) = f(t) for all values of t. The graph of an even function is always symmetrical about the vertical axis (that is, we have a mirror image through the y-axis). The waveforms shown below represent even functions: f(t) = 2 cos t Graph of f(t) = 2 cos(t), an even function. Notice that we have a mirror image through the `f(t)` axis. Even Square wave 12345-1-2-3-4-5123-1-2-3tf(t)Graph of an even step function. Triangular wave Graph of an even triangular function. In each case, we have a mirror image through the `f(t)` axis. Another way of saying this is that we have symmetry about the vertical axis. Odd Functions A function `y=f(t)` is said to be odd if `f(-t) = - f(t)` for all values of t. The graph of an odd function is always symmetrical about the origin. Origin Symmetry A graph has origin symmetry if we can fold it along the vertical axis, then along the horizontal axis, and it lays the graph onto itself. Another way of thinking about this is that the graph does exaclty the opposite thing on each side of the origin. If the graph is going up to the right on one side of the origin, then it will be going down to the left by the same amount on the other side of the origin. Examples of Odd Functions The waveforms shown below represent odd functions. Sine Curve y(x) = sin x 0.51.52-0.5--1.5-21-1xyGraph of y(x) = sin(x), an odd function. Notice that if we fold the curve along the y-axis, then along the t-axis, the graph maps onto itself. It has origin symmetry. "Saw tooth" wave Graph of a sawtooth function which is odd. Odd Square wave 12345-1-2-3-4-5123-1-2-3tf(t)Graph of an odd square wave. Each of these three curves is an odd function, and the graph demonstrates symmetry about the origin. Exercises Download graph paper Sketch each function and then determine whether each function is odd or even: (a) `f(t)={(e^t,text(if ) -pi le t lt 0),(e^-t,text(if ) 0 le t lt pi):}` Answer Graph of a split function. We can see from the graph that it is even. OR: The function is even since `f(-t) = f(t)` for all values of t. (b) `f(t)={(-1,text(if ) 0 le t lt pi/2),(1,text(if ) pi/2 le t lt (3pi)/2),(-1,text(if ) (3pi)/2 le t lt 2pi) :}` and f(t) = f(t + 2) (This last line means: Periodic with period = 2) Answer Graph of a step function. We can see from the graph that it is even. OR: The function is even since `f(-t) = f(t)` for all values of t. (c) `f(t)={(-t+pi,text(if ) -pi le t lt 0),(-t-pi,text(if ) 0 le t lt pi):}` Answer Graph of a split function. We can see from the graph that the function is odd. OR: The function is odd since `f(-t) = -f(t)` for all values of t. (d) `f(t)={(t-pi,text(if ) -pi le t lt 0),(-t+pi,text(if ) 0 le t lt pi):}` Answer Graph of a split function. We can see from the graph that it is neither odd nor even. (e) `f(t)={(t+pi,text(if ) -pi le t lt 0),(-t+pi,text(if ) 0 le t lt pi):}` Answer Graph of a split function. We can see from the graph that it is even. OR: The function is even since `f(-t) = f(t)` for all values of t. (f) `f(t)={((t+pi/2)^2,text(if ) -pi le t lt 0),(-(t-pi/2)^2,text(if ) 0 le t lt pi):}` Answer 0.5-0.5-12-1-2-3tf(t)Graph of a split function. We can see from the graph that the function is odd. OR: The function is odd since `f(-t) = -f(t)` for all values of t. Page 3 By D Hu and M Bourne Most functions you are familiar with are defined in the same manner for all values of x. However, there are some functions which are defined differently in different domains. These are known as split functions (or piecewise-defined functions). Because split functions may have drastically different behaviours in different domains (that is, for different x-values), it is quite common for a split function to be non-continuous (and as we learn later, it cannot be differentiated). Example 1 - Ordinary Function for Comparison f(x) = -x2 + 4 This function is not a split function. It is defined the same way for all values of x. To find the value of the function at a given x-value, simply substitute into f(x) = -x2 + 4 Some values for f(x) = -x2 + 4 are as follows: x -3 -2 -1 0 1 2 3 f(x) -5 0 3 4 3 0 -5 Example 2 - Split Function ` f(x)={(2x+3,text(for ) x=1):} ` In the region x < 1, we have a straight line with slope 2 and y-intercept `3`. As x approaches `1`, the value of the function approaches `5` (but does not reach it because of the "`

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