Mathematics homework help

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1. Let: 𝑓(𝑥) = −|5𝑥 − 2| and 𝑔(𝑥) = −3
a) SOLVE for x:
f (x) = g(x)
. What information about the graphs of these functions does the
solution give us?
b) SOLVE the inequality:
f (x)  g(x)
c) Graph the two functions on the same coordinate plane. Does the graph back up your solution from
part (b)?
2. Sketch the graph: 𝑓(𝑥) = (𝑥 − 2)
2
(𝑥 + 1)(𝑥 + 2)
3
3. a) Let 𝑓(𝑥) = 𝑥
4 − 𝑥
3 + 7𝑥
2 − 9𝑥 − 18 and
( ) 9
2
g x = x + . Use long division to show that g is a
factor of f. Explain.
b) Use the result found above to find all of the zeros (real and complex) of the polynomial f.
4. Let 𝑓(𝑥) = 4𝑥
4 + 5𝑥
3 + 9𝑥
2 + 10𝑥 + 2.
a) List all of the possible rational zeros.
b) Use the division to factor the polynomial
f (x)
completely. HINT: -1/4 is a zero of f .
c) Find all of the zeros of
f (x)
and state the multiplicity of each zero.
5. Let the function f be defined as follows:
𝑓(𝑥) =
𝑥
𝑥
2 − 4
a) What is the domain of f ?
b) Does the graph of f have any vertical asymptotes? If so, find them.
c) Does the graph of f have any horizontal asymptotes? If so, find them.
d) Where will the graph of f intercept the x-axis?
e) Where will the graph of f intercept the y-axis?
f) Sketch the graph of f.
6.) (Bonus Problem) Graph the function:
𝑓
(
𝑥
)
=
𝑥
2
−
5
𝑥
+
4
𝑥
2
−
2
𝑥
+
1