Deck 3: Linear, Quadratic, Polynomial, and Rational Functions

Question 1

Write the domain of the function r(x) as a union of intervals.
$r(x)=\frac{5 x^{7}-6 x^{5}-3}{x^{2}-5}$

A) $(-\infty,-\sqrt{5}) \cup(\sqrt{5}, \infty)$
B) $(-\infty,-\sqrt{5}) \cup(-\sqrt{5}, \sqrt{5})$
C) $(-\sqrt{5}, \sqrt{5})$
D) $(-\infty,-\sqrt{5}) \cup(-\sqrt{5}, \sqrt{5}) \cup(\sqrt{5}, \infty)$

Answer 1

$(-\infty,-\sqrt{5}) \cup(-\sqrt{5}, \sqrt{5}) \cup(\sqrt{5}, \infty)$

Question 2

Write the domain of the function r(x) as a union of intervals.
$r(x)=\frac{x^{9}-7 x^{5}-14}{x^{2}-7 x+49}$

Answer 2

$(-\infty, \infty)$

Question 3

Suppose $p(x)=\frac{9 x+7}{x^{2}+81}$ and $q(x)=\frac{x^{2}+9}{9 x-7}$ . Write the expression (p + q)(x) as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Answer 3

$\frac{x^{4}+171 x^{2}+680}{9 x^{3}-7 x^{2}+729 x-567}$

Question 4

Suppose $p(x)=\frac{9 x+7}{x^{2}+81}$ and $r(x)=\frac{9}{9 x^{2}+7}$ . Write the expression (p + r)(x) as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 5

Suppose $p(x)=\frac{5 x+3}{x^{2}+15}$ and $r(x)=\frac{5}{5 x^{2}+3}$ . Write the expression (3p - 15r)(x) as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 6

Suppose $p(x)=\frac{7 x+5}{x^{2}+49}$ and $q(x)=\frac{x^{2}+7}{7 x-5}$ . Write the expression (pq)(x) as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 7

Suppose $r(x)=\frac{5}{5 x^{2}+3}$ . Write the expression (r(x))² as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

A) $-\frac{25}{25 x^{4}+30 x^{2}+9}$
B) $\frac{25}{25 x^{4}+30 x^{2}+9}$
C) $\frac{25}{25 x^{4}-30 x^{2}+9}$
D) $-\frac{25}{25 x^{4}-30 x^{2}+9}$

Question 8

Suppose $p(x)=\frac{3 x+1}{x^{2}+9}$ and $q(x)=\frac{x^{2}+3}{3 x-1}$ . Write the expression (q(x))² p(x) as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 9

Suppose $q(x)=\frac{x^{2}+7}{7 x-5}$ and $r(x)=\frac{7}{7 x^{2}+5}$ . Write the expression $(r \circ q)(x)$ as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 10

Suppose $p(x)=\frac{4 x+2}{x^{2}+16}$ and $q(x)=\frac{x^{2}+4}{4 x-2}$ . Write the expression $(p \circ q)(x)$ as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 11

Suppose $r(x)=\frac{3}{3 x^{2}+1}$ . Write the expression $\frac{r(3+x)-r(3)}{x}$ as a ratio, with the numerator and denominator each written as a sum of terms of the form cx^m.

Question 12

What is the domain of $f(x)=\frac{x+8}{x^{2}+9}$ ?

A) $(-\infty,-9) \cup(-9, \infty)$
B) $(-\infty,-9) \cap(-9, \infty)$
C) $(-\infty, \infty)$
D) $(-\infty, 9) \cup(9, \infty)$

Question 13

Find two distinct numbers x such that $t(x)=\frac{1}{2}$ , where $t(x)=\frac{x+17}{x^{2}+16}$ .

Question 14

What is the range of s, where s is defined by $s(x)=\frac{x+3}{x^{2}+9}$ ?

Question 15

Write the expression $\frac{3 x+1}{x-9}$ in the form $G(x)+\frac{R(x)}{q(x)}$ , where q is the denominator of the given expression and G and R are polynomials with deg R < deg q.

Question 16

Write the expression $\frac{x^{2}}{x+9}$ in the form $G(x)+\frac{R(x)}{q(x)}$ , where q is the denominator of the given expression and G and R are polynomials with deg R < deg q.

Question 17

Write the expression $\frac{x^{6}+2 x^{3}+8}{x^{2}-2 x-8}$ in the form $G(x)+\frac{R(x)}{q(x)}$ , where q is the denominator of the given expression and G and R are polynomials with deg R < deg q.

Question 18

Find a constant c such that $f\left(10^{100}\right) \approx 4$ , where $f(x)=\frac{x^{3}-20 x^{2}+12 x-16}{4 x^{3}-8 x^{2}+x-3}$ .

A) 4
B) 16
C) 0
D) -4

Question 19

Find the asymptotes of the graph of the function $r(x)=\frac{6 x^{4}-x^{2}-3}{x^{4}+6 x^{2}+6}$ .

A) y = 6
B) y = 0
C) x = 6
D) x = 0

Question 20

Find the asymptotes of the graph of the function
$r(x)=\frac{x+8}{x^{2}-17 x+72}$ .

Question 21

Suppose p(x) = x² + 6x + 4, q(x) = 6x³ - 4x + 1. Write the expression (p + q)(x) as a sum of terms, each of which is a constant times a power of x.

A) 6x³ + x² + 2x + 5
B) 6x³ + x² + 2x - 5
C) 6x³ + x² - 2x + 5
D) 6x³ + x² - 2x - 5

Question 22

Suppose p(x) = x² + 9x + 7, r(x) = 18x³ + 9. Write the expression (9p - 7r)(x) as a sum of terms, each of which is a constant times a power of x.

Question 23

Suppose p(x) = x² + 3x + 1, r(x) = 6x³ + 3. Write the expression (p(x))²(r(x) - 3) as a sum of terms, each of which is a constant times a power of x.

Question 24

Suppose q(x) = 5x³ - 3x + 1, r(x) = 10x³ + 5. Write the expression $(q \circ r)(x)$ as a sum of terms, each of which is a constant times a power of x.

Question 25

Suppose p(x) = x² + 4x + 2, q(x) = 4x³ - 2x + 1, r(x) = 8x³ + 4. Write the expression $((p+q) \circ r)(x)$ as a sum of terms, each of which is a constant times a power of x.

Question 26

Suppose p(x) = x² + 8x + 6. Write the expression $\frac{p(h+1)-p(1)}{h}$ As a sum of terms, each of which is a constant times a power of h.

A) 10 - h
B) 10 + h
C) -6 + h
D) -6 - h

Question 27

Find all real numbers x such that
x⁴ + 36x² - 2,592 = 0

Question 28

Find all real numbers x such that: $x^{6}-63 x^{3}-64=0$

A) 4, 1
B) -4, 1
C) 4, -1
D) -4, -1

Question 29

Find all real numbers x such that:
x⁴ - 53x² + 196 = 0

Question 30

Suppose q(x) = $x^3 - 49x + b$ . The point (7, b) lies on the graph of q.

Question 31

is $x^8 - 1 = (x^4 + 1)(x^2 + 1)(x + 1)(x - 1)$ correct?

Question 32

Find a number b such that 6 is a zero of the polynomial p defined by p(x) = -36 + bx - 6x² + x³.

A) 36
B) -36
C) 6
D) -6

Question 33

Find a number b such that 1 is a zero of the polynomial p defined by p(x) = b - 9x + bx² + 9x³.

Question 34

Find a polynomial p of degree 3 such that -6, 6, and 7 are zeros of p and p(0) = 36. Write your answer in descending order by degree.

Question 35

Find a polynomial p of degree 3 such that -1, $\frac{1}{8}$ , and 1 are zeros of p and p(0) = $\frac{1}{8}$ . Write your answer in descending order by degree.

Question 36

If p(x) = a²x² + 4x + 5 and q(x) = (1 - 2a)x² - 8x - 5, find a real number a such that (p + q)(x) has degree 1.

A) 4
B) -4
C) 0
D) 1

Question 37

Suppose d is a real number. Then $(d + 6)^4 = d^4 + 6^4$ if and only if d = 0

Question 38

The only possible integer zeros of $p(x) = 2x^5 - bx^3 + 4x - 2$ are 1 and 2.

Question 39

Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Question 40

Find all choices of (b, c, d) such that 9 and -9 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Question 41

Factor x²⁴ - y¹² as nicely as possible.

Question 42

Evaluate the expression $3^{2}-2^{3}$ .

Question 43

Write $27^{5} \cdot 9^{1}$ as a power of 3.

Question 44

Simplify the given expression by writing it as a power of a single variable. $x^{7}\left(x^{5}\right)^{2}$

Question 45

Simplify the given expression by writing it as a power of a single variable. $w^{2}\left(w^{3}\right)^{1}$

Question 46

Simplify the given expression by writing it as a power of a single variable. $t^{9}\left(t^{8}\left(t^{-2}\right)^{11}\right)^{3}$

Question 47

Write $\frac{8^{400}}{2^{4}}$ as a power of 2.

Question 48

Find integers m and n such that $11^{m} \cdot 7^{n}=456,533$ .

Question 49

Simplify the given expression. $\frac{\left(x^{8}\right)^{9} y^{24}}{x^{6}\left(y^{9}\right)^{2}}$

Question 50

Simplify the given expression. $\frac{\left(x^{3} y^{6}\right)^{4}}{\left(x^{7} y^{-3}\right)^{6}}$

Question 51

Simplify the given expression. $\left(\frac{\left(x^{-5} y^{9}\right)^{-8}}{\left(x^{-9} y^{-4}\right)^{-5}}\right)^{-4}$

Question 52

Find a formula for $f \circ g$ if $f(x)=x^{4}$ and $g(x)=x^{6}$ .

Question 53

Find a formula for $f \circ g$ if $f(x)=6 x^{12}$ and $g(x)=3 x^{5}$ .

Question 54

Find the function which is graphed below

A) $f(x)=\frac{1}{x^{2}}+8$
B) $f(x)=\frac{1}{x^{2}}-8$
C) $f(x)=-\frac{1}{x^{2}}+8$
D) $f(x)=-\frac{1}{x^{2}}-8$

Question 55

Find the function which is graphed below.

A) $f(x)=\frac{1}{x^{2}}+5$
B) $f(x)=\frac{1}{x^{2}}-5$
C) $f(x)=-\frac{1}{x^{2}}+5$
D) $f(x)=-\frac{1}{x^{2}}-5$

Question 56

Find the function which is graphed below.

A) $f(x)=\frac{1}{x}+5$
B) $f(x)=\frac{1}{x}-5$
C) $f(x)=-\frac{1}{x}+5$
D) $f(x)=-\frac{1}{x}-5$

Question 57

Find the function which is graphed below.

A) $f(x)=\frac{3}{x}+6$
B) $f(x)=\frac{3}{x}-6$
C) $f(x)=-\frac{3}{x}+6$
D) $f(x)=-\frac{3}{x}-6$

Question 58

Find the function which is graphed below.

A) $f(x)=\frac{1}{x}+9$
B) $f(x)=\frac{1}{x}-9$
C) $f(x)=-\frac{1}{x}+9$
D) $f(x)=-\frac{1}{x}-9$

Question 59

Evaluate the indicated quantity.25^3/2

A) 25
B) 5
C) 37.5
D) 125

Question 60

Evaluate the indicated quantity.(-512)^5/3

A) 32,768
B) -32,768
C) 512
D) Undefined

Question 61

Expand the indicated expression. $(3+\sqrt{3})^{2}$

Question 62

Expand the indicated expression. $(5-\sqrt{3})^{4}$

Question 63

Expand the indicated expression. $(10-\sqrt{x})^{2}$

A) $100+x-20 \sqrt{x}$
B) $100+x+20 \sqrt{x}$
C) $100+x^{2}-20 x$
D) $100+x^{2}+20 x$

Question 64

Expand the indicated expression. $(1+2 \sqrt{5 x})^{2}$

A) $1+20 x+2 \sqrt{5 x}$
B) $1+20 x+\sqrt{5 x}$
C) $1+20 x+4 \sqrt{5 x}$
D) $1-20 x+4 \sqrt{5 x}$

Question 65

Find a formula for the inverse function $f^{-1}$ of the indicated function. $f(x)=x^{12}$

Question 66

Find a formula for the inverse function $f^{-1}$ Of the indicated function. $f(x)=x^{\frac{1}{11}}$

A) $f^{-1}(x)=\frac{1}{x^{\frac{1}{11}}}$
B) $f^{-1}(x)=\frac{1}{x^{11}}$
C) $f^{-1}(x)=x^{\frac{1}{11}}$
D) $f^{-1}(x)=x^{11}$

Question 67

Find a formula for the inverse function $f^{-1}$ of the indicated function. $f(x)=2-x^{7}$

A) $f^{-1}(x)=(2-x)^{\frac{1}{7}}$
B) $f^{-1}(x)=2-x^{\frac{1}{7}}$
C) $f^{-1}(x)=2+x^{\frac{1}{7}}$
D) $f^{-1}(x)=(2+x)^{\frac{1}{7}}$

Question 68

Find a formula for the inverse function $f^{-1}$ of the indicated function. $f(x)=6 x^{\frac{4}{5}}+3$

Question 69

Find a formula for (f $\circ$ g)(x) assuming that f and g are the indicated functions. $f(x)=x^{\frac{3}{4}}$ and $g(x)=x^{\frac{4}{11}}$ .

Question 70

Find a formula for (f $\circ$ g)(x) assuming that f and g are the indicated functions. $f(x)=x^{\frac{2}{3}}$ and $g(x)=x^{\frac{4}{5}}$ .

A) $(f \circ g)(x)=x^{\frac{15}{8}}$
B) $(f \circ g)(x)=x^{\frac{8}{15}}$
C) $(f \circ g)(x)=x^{\frac{2}{5}}$
D) $(f \circ g)(x)=x^{\frac{4}{3}}$

Question 71

Find all real numbers x such that satisfy the indicated equation. $x+26=15 \sqrt{x}$

Question 72

Find all real numbers x such that satisfy the indicated equation. $x-2 \sqrt{x}=48$

A) $x=64$ And $x=-36$
B) $x=64$ and $x=36$
C) $x=-64$
D) $x=64$

Question 73

Find all real numbers x such that satisfy the indicated equation. $x^{\frac{1}{2}}-16 x^{\frac{1}{4}}+63=0$

Question 74

Suppose x is a number such that 2^x = 5. Evaluate 16^x.

A) 625
B) 20
C) 80
D) 40

Question 75

Evaluate the $(f \circ g)\left(\frac{1}{4}\right)$ assuming that f and g are the functions defined by f(x) = 3^x and $g(x)=\frac{x+1}{x}$ .

Question 76

What is the domain of the function $(3+x)^{\frac{1}{6}}$ ? Write your answer using set builder notation.

Question 77

If x and y are irrational numbers, then xy is always an irrational number.

Question 78

If x and y are irrational numbers, then x + y may be a rational number.

Question 79

Find the vertex of the graph of the function
$f(x)=5 x^{2}+1$ .

Question 80

Find the vertex of the graph of the function f(x) = (x - 11)² + 9.