Math 3 Group Work Solutions, 1/6
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1. Let f (x) = x and g(x) = 4 − x2 . What is f ◦ g? What are the domain and
range of f ◦ g?
Solution:
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f ◦ g(x) = f (g(x)) = f (4 − x2 ) = 4 − x2 .
The domain of f ◦ g is all values of x such that 4 − x2 ≥ 0, or x2 ≤ 4.
Thus, the domain is [−2, 2].
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The range of f ◦ g is all values of 4 − x2 , where −2 ≤ x ≤ 2. Thus, the range is [0, 2].
2. Let f (x) = x2 and g(x) =
√
1 − x2 . Find the domain and ranges of g and f ◦ g.
Solution:
The domain of g is all values of x such that 1 − x2 ≥ 0, so x2 ≤ 1.
Thus, the domain is [−1, 1]. √
The range of g is all values of 1 − x2 , where −1 ≤ x ≤ 1. Thus, the range is [0, 1].
√
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First, f ◦ g(x) = f (g(x)) = f ( 1 − x2 ) = ( 1 − x2 )2 = 1 − x2 .
The domain of f ◦ g is all values of x in the domain of g.
Thus, the domain of f ◦ g is [−1, 1].
The range of f ◦ g is all values of 1 − x2 , where −1 ≤ x ≤ 1. Thus, the range is [0, 1]
*****Be careful here! It is tempting to say that the domain of f ◦ g is (−∞, ∞) because
that is the domain of 1 − x2 ; however when we look at the domain of f ◦ g we must only
include elements in the domain of g. *****
3. Are the following functions even or odd or neither?
Solution:
f (x) is neither: It is not symmetric about the y − axis.
g(x) is even because it is symmetric about the y-axis: g(x) = g(−x).
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4. Is the function f (x) = 3x9 − 3x7 odd, even, or neither? Can you generalize this
to the sum of any two odd functions?
Solution:
Check: f (−x) = 3(−x)9 − 3(−x)7 = −3x9 + 3x7 = −f (x). Thus, f (x) is odd.
For any two odd functions f and g, we have (f +g)(−x) = f (−x)+g(−x) = −f (x)−g(x) =
−(f + g)(x).
Thus, the sum of two odd functions is odd.
5. Suppose f is an even function and g is an odd function.
i. Is f g even, odd, or neither?
ii. Is f ◦ g even, odd, or neither? What about g ◦ f ?
How would your answers change if f and g are both odd functions?
Solution:
i. Check: f g(−x) = f (−x)g(−x) = f (x)(−g(x)) = −f (x)g(x). Thus, f g is odd.
ii. Check: f ◦ g(−x) = f (−g(x)) = f (g(x)) since f is even, so f ◦ g is even.
Check: g ◦ f (−x) = g(f (−x)) = g(f (x)) since f is even, so g ◦ f is even.
6. Classify the following functions as a polynomial (state its degree), rational
function, algebraic function, trigonometric function, exponential function, or
logarithmic function.
i. f (x) = log2 (x)
2x3
ii. g(x) = 1−x
2
√
iii. h(x) =
x3√
−1
1− 3 x
Solution:
i. Logarithmic function
ii. Rational function
iii. Algebraic function
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