1. Find the natural domain of the function f(x) =...

1. Find the natural domain of the function $f(x) = \frac{\sqrt{3-x}}{x}$. A. $(-\infty, -3]$ B. $[3, \infty)$ C. $(0, \infty)$ D. $(-\infty, 0) \cup (0, 3]$ E. $(-\infty, 0) \cup (0, 3)$ 2. Let $f(x) = x^2 - 2x$. Evaluate and simplify $f(x+h)$. A. $x^2 + 2xh + h^2 - 2x - 2h$ B. $x^2 - 2x + h$ C. $x^2 - 2x + h^2 - 2h$ D. $x^2 + h^2$ E. $x^2 + 2xh + h^2 - 2x + 2h$ 3. In which one of the following equations is $y$ a function of $x$? A. $x^4 - 2y^3 = 9$ B. $3|x| + 5|y| = x$ C. $2x^5 - 5y^4 = 7$ D. $x^2 + y^2 = 1$ E. $2x^3 + |y| = 3$ 4. Which one of the following statements is true for the function $f(x) = \frac{5x^2 - 9}{x^3}$? A. $f$ is neither an even function nor an odd function. B. $f$ is an even function. C. $f$ is an odd function. D. $f$ is both an even function and an odd function. E. None of these statements are true.

Transcript

00:01 Hello everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 61 in chapter 1, section 1 of calculus early transcendentals.

00:11 Now this problem provides us with nine different claims about functions and ask us to verify whether or not these statements are true with an explanation or false with a counter example.

00:23 Let's start with part a.

00:25 This part gives us a function f of x equals 2x minus 38 and states that the range of this function is the entire real number line.

00:35 This is true.

00:41 Set f of x equal to some number a.

00:46 Then this means that a equals 2x minus 38.

00:53 And that means that we may find an x that is equal to a plus 38 over two.

01:04 Notice that any number a can be plugged in here and there will be an input x associated to it.

01:10 And therefore, the range of this function is every single number.

01:17 Now let's go on to part b.

01:20 This statement says that the expression y equals x to the six plus one is not a function because plugging negative one or one into this expression for x results in the same number two for y this is false and this is because functions only require a unique output for every input but they do not require unique input for every output so we are indeed allowed to have two different inputs produce the same output and have it still be a function.

02:01 Now let's go on to part c.

02:05 This statement says that if we have f of x equal to x to the negative first power, then plugging 1 over x into f is the same thing as 1 divided by f of x.

02:18 This is true.

02:22 Notice that f of 1 over x is equal to 1 over x to the negative first power.

02:32 Now 1 to the negative first is just 1, and that is over x to the negative first power.

02:39 And that is just 1 over f of x.

02:45 Now let's go on to part d.

02:48 The statement says that f of f of x is always equal to the square of f of x for any function f.

02:57 This is false...

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