If f(x)=-√(x), can f(x) ever be a) greater than 0 , b) equal...

Key Concepts

Inequalities in Functions

Analyzing whether a function's output meets specific inequality conditions involves examining the algebraic structure of the function. Recognizing properties such as the nonnegativity of the square root and the effects of multiplying by -1 helps determine if the function can be positive, zero, or negative.

Domain and Range Analysis

Understanding the domain (all valid input values) and range (all possible output values) of a function is crucial for determining its behavior. The limitations imposed by the square root function and subsequent transformations help in analyzing whether the function's output can be greater than, equal to, or less than a given value.

Square Root Function

The square root function takes a nonnegative input and returns a nonnegative output. This fundamental property means that for any valid input in its domain, the result of a square root is always zero or positive.

Function Transformations (Reflection)

When a function is multiplied by -1, the graph of the original function is reflected across the horizontal axis. This means that nonnegative outputs become nonpositive, altering the range of the function accordingly.

Transcript

00:03 Okay, here we have f of x is equal to x times the square root of x, and we need to find f prime of x.

00:10 So we will try to do this by finding the limit as each approach is zero of the difference quotient, f of x plus h, minus f of x over h, and we'll take that limit as h approach is zero.

00:47 So, f prime of x will be delimited as h approaches zero of this difference quotient.

00:53 To take the limit as each approach is zero.

01:02 F of x plus h will be x plus h times the square root of x plus h.

01:19 And then we need to subtract f of x, which is x times the square root of x.

01:30 And that all has to be put over h because we need to take the limit of this difference quotient as age approaches zero.

01:47 Our next step would be to multiple.

01:52 The numerator and the denominator by the conjugate of this numerator.

02:01 So we're going to multiply the numerator by x plus h, square root of x plus h, plus the square root of x, root x, and we're going to multiply the bottom by the same x plus h, square root of x plus h, plus, plus, plus x -rude x.

03:09 So this numerator is going to be multiplied by its conjugate x plus h square root of x plus h plus x -rude x -x and the denominator h needs to be multiplied by the same thing.

03:23 So this original fraction, this original expression is being multiplied by this expression over itself.

03:32 So basically it's being multiplied by one so it doesn't change the value of the original expression.

03:37 Now, as complicated looking as this is, you got two terms here, multiplying two terms here, and we're going to use oil.

03:46 So this first term is going to times this first term, and then for the outside terms, this term here will multiply a positive x root x.

03:59 And then for the inside term, this term here will multiply a negative x root x.

04:06 So those will cancel.

04:07 That's what makes things a little bit easier.

04:08 And then the last term in this bracket times the last term in this bracket, negative x root x times positive x root x.

04:17 And the denominator we simply have h multiplying all this, which we will probably just rewrite it as such, because we hopefully will have an h somewhere up top here.

04:26 That could be canceled with the h down here.

04:37 So as messy and complex looking as this was, once you start multiple, applying these two terms times these two terms.

04:49 It works out rather nicely.

04:52 So x plus h, root x plus h times x plus eight, root x plus h, x plus h times x plus h, x plus h to the second.

05:02 Square root of x plus h times square of x plus h, simply x plus h.

05:06 So we're really going to have an x plus h to the third power here...

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