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Functions : Odd, Even, One-to-one, Domain, Range and Function Composition

Practice Problems

Compare the graph of the given quadratic function f with the graph of y = x2.
1) f(x) = (x - 2)2 + 3

Determine if the function is even, odd, or neither.
2) f(x) = 2x5 + 2x3

Decide whether the relation defines a function.
3) {(-8, 2), (-8, 8), (-1, 8), (5, 6), (8, 7)}

5) y2 = 3x

Find the domain and range of the inverse of the given function.
7) f(x) = -4x - 2

Compute and simplify the difference quotient
8) f(x) = 7x - 9

Consider the function h as defined. Find functions f and g so that (f ∘ g)(x) = h(x).
9) h(x) =

Find the requested value.
10) f(3) for f(x) = 3x + 1, if x < 1
f(3) for f(x) = 3x, if 3 &#8804; x &#8804; 8
f(3) for f(x) = 3 - 5x, if x > 8

Determine whether the function is symmetric with respect to the y-axis, symmetric with respect to the x-axis, symmetric with respect to the origin, or none of these.
11) f(x) = -5x3 + 2x

If the following defines a one-to-one function, find its inverse. If not, write "Not one-to-one."
12) {(-2, 4), (-1, 4), (0, 1), (1, -5)}

If f is one-to-one, find an equation for its inverse.
13) f(x) = x3 - 5

Perform the requested operation or operations.
14) f(x) = , g(x) = 8x - 14
Find (f &#8728; g)(x).

Graph the function.
16) f(x) = 4x + 2 if x < -2
f(x) = x if -2 &#8804; x &#8804; 3
f(x) = 3x-1 if x > 3

Find the indicated composite for the pair of functions.
18) (g &#8728; f)(x): f(x) = , g(x) = 6x + 3

Give the domain and range of the relation.
20) y = (x + 4)2 - 7

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Practice Problems

Compare the graph of the given quadratic function f with the graph of y = x2.
1) f(x) = (x - 2)2 + 3

First expand this function:
f(x) = (x - 2)(x - 2) + 3
= x2 - 4x - 4 + 3
= x2 - 4x - 1

If we compare these two function we see that the difference is in the (-4x - 1) term at the end. The x2 is the same. So basically what does this end term do to the function?

The -4x - 1 term shifts the graph to the right and down, due to the negative signs before each term.
You can use the following site to compare: http://people.hofstra.edu/steven_r_costenoble/Graf/Graf.html

Determine if the function is even, odd, or neither.
2) f(x) = 2x5 + 2x3

To do this, you take the function and plug -x in for x, and then simplify. If you end up with the exact same function that you started with (that is, if f(-x) = f(x)), then the function is even. If you end up with the exact opposite of what you started with (that is, if f(-x) = -f(x)), then the function is odd.

Plug in -x for x
f(-x) = 2(-x)5 + 2(-x)3
= - 2x5 - 2x3

This is exactly opposite of what we started with so this function is odd!

Decide whether the relation defines a function.
3) {(-8, 2), (-8, 8), (-1, 8), (5, 6), (8, 7)}

The easiest way to determine this is to look at the first number in each pair. If any of them are the same, then this is a relation, not a function. ...

Solution Summary

Odd, Even, One-to-one functions, Domain, Range and Function Composition are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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