Answers
g(x) as given has no inverse because there are instances of two x values giving the same value of g(x). For instance,
x = -1 ⇒ g(-1) = 4 (-1 + 3)² - 8 = 8
x = -5 ⇒ g(-5) = 4 (-5 + 3)² - 8 = 8
Only a one-to-one function can have an inverse. g(x) is not one-to-one.
However, if we restrict the domain of g(x), we can find an inverse over that domain. Let [tex]g^{-1}(x)[/tex] be the inverse of g(x). Then by definition of inverse function,
[tex]g\left(g^{-1}(x)\right) = 4 \left(g^{-1}(x) + 3\right)^2 - 8 = x[/tex]
Solve for the inverse:
[tex]4 \left(g^{-1}(x) + 3\right)^2 - 8 = x[/tex]
[tex]4 \left(g^{-1}(x) + 3\right)^2 = x + 8[/tex]
[tex]\left(g^{-1}(x) + 3\right)^2 = \dfrac{x + 8}4[/tex]
[tex]\sqrt{\left(g^{-1}(x) + 3\right)^2} = \sqrt{\dfrac{x + 8}4}[/tex]
[tex]\left| g^{-1}(x) + 3 \right| = \dfrac{\sqrt{x+8}}2[/tex]
Recall the definition of absolute value:
[tex]|x| = \begin{cases}x & \text{if }x\ge0\\-x&\text{if }x<0\end{cases}[/tex]
This means there are two possible solutions for the inverse of g(x) :
• if [tex]g^{-1}(x) + 3 \ge 0[/tex], then
[tex]g^{-1}(x) + 3 = \dfrac{\sqrt{x+8}}2 \implies g^{-1}(x) = -3+\dfrac{\sqrt{x+8}}2[/tex]
• otherwise, if [tex]g^{-1}(x)+3<0[/tex], then
[tex]-\left(g^{-1}(x) + 3\right) = \dfrac{\sqrt{x+8}}2 \implies g^{-1}(x) = -3-\dfrac{\sqrt{x+8}}2[/tex]
Which we choose as the inverse depends on how we restrict the domain of g(x). For example:
Remember that the inverse must satisfy
[tex]g\left(g^{-1}(x)\right) = x[/tex]
In the first case above, [tex]g^{-1}(x) + 3 \ge 0[/tex], or [tex]g^{-1}(x) \ge -3[/tex]. This suggests that we could restrict the domain of g(x) to be [tex]x \ge -3[/tex].
Then as long as [tex]x \ge -3[/tex], the inverse is
[tex]g^{-1}(x) = -3+\dfrac{\sqrt{x+8}}2[/tex]
Related Questions
solve the equation (x-4) (x+9) = 0
Answers
Answer:
x = 4,-9
Step-by-step explanation:
This is a quadratic equation in x-intercept pattern. Since this is completely factored, we can simply solve the equation like a linear.
First, separate two equations. Let r be the roots of equation, therefore:
[tex]\displaystyle \large{r=\begin{cases}x-4=0\\ x+9=0 \end{cases}}[/tex]
I use the r-variable to denote the roots of equations (not necessary), simply separate both then solve it like a linear.
[tex]\displaystyle \large{r=\begin{cases}x-4+4=0+4\\ x+9-9=0-9 \end{cases}}\\\displaystyle \large{r=\begin{cases}x=4\\ x=-9 \end{cases}}[/tex]
Therefore, the roots of equation are x = 4 or x = -9. In short, we can write as x = 4,-9.
The reason why we use OR instead of AND because one of x-values satisfy the quadratic equation hence ‘or’ is more suitable than ‘and’. However, you must write all valid solutions instead of writing only one x-value from two.
___________________
Let me know in the comment if you have any questions.
What do the following two equations represent?
. 4x + 2y = 10
• 4x – 2y = –10
Choose 1 answer:
The same line
Distinct parallel lines
Perpendicular lines
Intersecting, but not perpendicular lines
Answers
Answer:
Intersecting but not perpendicular
Step-by-step explanation:
2y=-4x+10= Y=-2x+5
-2y=-4x-10= Y=2x+5
jack has $37 and saves $6.50 per week diane has $9 and saves $8.25 per week write and solve an equation to find out how many weeks it would take for jack and diane to have saved the same amount of money
Answers
-1.75x = 28
X= 16
express the following fractions as the sum or difference of two fractions (x^2+y^2)/x^4
Answers
Answer:
[tex]\frac{1}{x^2}[/tex] + [tex]\frac{y^2}{x^4}[/tex]
Step-by-step explanation:
[tex]\frac{x^2 + y^2}{x^4}[/tex] = [tex]\frac{x^2}{x^4}[/tex] + [tex]\frac{y^2}{x^4}[/tex] = [tex]\frac{1}{x^2}[/tex] + [tex]\frac{y^2}{x^4}[/tex]
The solution of the expression (x²+y²)/x⁴ is [(1/x² + y²/x⁴)].
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
The given expression is (x²+y²)/x⁴. The expression will be solved as,
E = (x²+y²)/x⁴
Separate the terms into two fractions,
E = (x²+y²)/x⁴
E = (x²/x⁴) + ( y² / x⁴)
Divide the divisible terms and solve,
E = (1/x² + y²/x⁴)
To know more about an expression follow
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What is the area, in square centimeters, of the trapezoid below?
Answers
Answer:
36 cm²
Step-by-step explanation:
1/2 × (8.9+5.5) ×5.5
1/2×14.4×5.5
1/2×72
36 cm²
Find the solutions to x2 = 24.
A. x = 16.2
X
B. X= 14,6
C. X= +6.14
D. X = +216
Answers
QUICKEST ANSWER GETS BRAINLEIST.
Answers
15a. Cindy can make $100 in four days babysitting. At that rate, how many day will it take
her to make $425.
Answers
Answer:
4.2
Step-by-step explanation:
I have no idea how to do this. Please help.
Answers
Answer:
Area is 5 cm and perimeter is 12 cm
Step-by-step explanation:
Area: top left sqare:
1cm * 1cm = 1 cm
bottom rectangle:
1cm * 3 cm = 3 cm
top right square:
1cm * 1cm = 1 cm
Perimeter:
1 cm + 2 cm+ 3 cm+ 2 cm+ 1 cm + 1 cm + 1cm + 1cm = 12 cm
which ratio is equivalent to the ratio 7:6
Answers
Answer:
B. 21:18
Step-by-step explanation:
7/6 x 3 = 21/18
This means they are equivalent.
Answer:
Option B is correct.
Step-by-step explanation:
Let's first turn 7:6 into a fraction:
=> 7:6 = 7/6
Now, let's simplify all the options to see which one is the correct answer.
Option A:
=> 14:18=> 14/18=> 7/9=> 7/9 ≠ 7/6Therefore, Option A is incorrect.
Option B:
=> 21:18=> 21/18=> 7/6=> 7/6 = 7/6Therefore, Option B is correct.
Option C:
=> 21:12=> 21/12=> 7/4=> 7/4 ≠ 7/6Therefore, Option C is incorrect.
Option D:
=> 18:14=> 18/14=> 9/7=> 9/7 ≠ 7/6Therefore, Option D is incorrect.
We can conclude that Option B is correct.
[tex]BrainiacUser1357[/tex]
What is the common denominator of StartFraction 1 Over a EndFraction StartFraction 1 Over b EndFraction in the complex fraction StartFraction 1 Over a EndFraction minus StartFraction 1 Over b EndFraction divided by StartFraction 1 Over a EndFraction 1 Over b EndFraction?.
Answers
The common denominator is 1
Given the expression:
[tex]\frac{1/a-1/b}{1/a1/b}[/tex]Find the LCM of the numerator to have:
[tex]\dfrac{\frac{b-a}{ab} }{\frac{1}{a} \frac{1}{b} }[/tex]Simplify both the numerator and the denominator to have:
[tex]= \dfrac{\frac{b-a}{ab} }{\frac{1}{ab} }\\= \frac{b-a}{ab} \times ab\\=\frac{b-a}{1} \times 1 \\=\frac{b-a}{1}[/tex]
From the result, we can see that the common denominator is 1
Learn more about fractions here: https://brainly.com/question/17743912
Answer:
here is the correct answer for edge2022
Step-by-step explanation:
The graph of y = x^2 -2x -3
a) Write down the coordinates of the turning point on the graph of y = x² – 2x - 3
(1)
(b) Use the graph to find the roots of the equation r? - 2x - 3 = 0
Answers
Answer:
See below
Step-by-step explanation:
A) Recall that a turning point is where the function changes from increasing to decreasing or vice versa. Since the graph of the function is a parabola, it only has 1 turning point, which is the vertex, or (1,-4). Therefore, the coordinates of the turning point are (1,-4).
B) The roots of the equation can be found by determining where the graph intercepts the x-axis (aka. x-intercepts). In this case, these are points (-1,0) and (3,0).
how do you solve 9r=3r+6
Answers
Subtract 3r from both sides
6r=6
Divide both sides by 6
r=1
Answer:
r=1
Step-by-step explanation:
9r=3r+6
-3r -3r
6r=6
/6 /6
r=1
Check your answer:
9(1)=3(1)+6
9=9 ✔️
Joe started to save money to purchase a new game console. His savings at the end of the first week included a jar of 468 nickels and dimes. The total value of the coins was $34.40. Which system of equations below would be the best to use to find the number of nickels and dimes Anthony had at the end of the first week?
Answers
Answer:
0.05x + 0.10y = 34.40
x + y = 468
Step-by-step explanation:
Step-by-step explanation:
The The guy above is correct
What's the percentage multipier for 0.6%?
Answers
Answer:
Solution Given;
0.6%=0.6/100=0.006
So
0.006 is the percentage multipier for 0.6%.
Let's check
We need to convert it into decimals
0.6%0.6×1/1000.6×0.0010.0060.006 is the answer
17) The growth of a certain bacteria colony is modeled by the function y = 100(1.20)*. Which statements are
correct?
A)
B)
100 is the initial number of bacteria.
x is the growth factor.
This is an exponential function.
1.20 is the number of time intervals.
D)
E)
y equals the final number of bacteria.
C)
Answers
Answer:
the corect answers is d and c
Step-by-step explanation:
Answer: B D E
Step-by-step explanation: I got it right on usetestprep
Suppose line g is the line with equation y=x. Given A(6,-2), B(6,3), and C(-2,-6), what are the coordinates of the vertices of A'B'C' for the reflection?
NEED ANSWER ASAP
Rg
Answers
Answer:
A'(-2, 6)B'(3, 6)C'(-6, -2)Step-by-step explanation:
Reflection across the line y=x swaps the x and y coordinates:
(x, y) ⇒ (y, x)
The image points are ...
A(6, -2) ⇒ A'(-2, 6)
B( 6, 3) ⇒ B'(3, 6)
C(-2, -6) ⇒ C'(-6, -2)
Which of the following is NOT a true statement?
pls answer i'll give brainliest
Answers
Answer:
C.) AFD measures 30°
Step-by-step explanation:
it is 150 degrees
Answer:
the answer is C
Step-by-step explanation:
A) angle EFC is 80 degrees so a is a true statement
B) angle BFC is 60 degrees and angle DFE is 30 degrees 60+30=90 so that's a true statement
D) angle AFB is 40 degrees and angle CFD is 50 degrees 40+50=90 so that a true statement
that leaves C and angle AFD is more than 30 degrees its 150 degrees
quick pls!! my math grade is at a B! I need to raise it.
Answers
-hello10olleh (hello)
Answers
Answer: The slope of the line is b. 3
Answer:b
Step-by-step explanation:
i need help assap
Consider the function g(x) shown below over a domain of [1,∞).
g(x) = 2(x – 1)2 – 3
Part A
Determine g–1(x).
Part B
Identify the domain and range of g(x) and g–1(x). Represent the domains and ranges in inequality, interval, and set notation. Describe the relationship between the domains and ranges of g(x) and g–1(x).
Part C
Graph g(x) and g–1(x). Describe the relationship between the graphs of g(x) and g–1(x).
Answers
Examine the function f(x)=log2x.
Which point is the intersection of f(x) and the line y=1?
(1,2)
(12,1)
(2,1)
(1,1)
Answers
Answer:
Is the question worded properly?
I find the intersection of the two lines to be (5,1), which is not an answer option.
Step-by-step explanation:
We want the value of x that will produce a value of 1, which will be the point the curve crosses y = 1.
log (10) = 1
Therefore 2x = 1, and x = 5
(5,1)
Alebgra!!
Please help me fast now plzzzzzzzzzzzzzzzzzzzzzzzzzzzzz i will give you alot of my points.
Question:
Find the perimeter of the rectangle with length (2w + 7) feet and width w feet.
Answers
Answer:
I think it's 18
Step-by-step explanation:
P=2(l+w)=2(7+2)=18
it's been awhile I believe this is it
A rectangle has four sides=2L+2W
2(2w+7)+2w
4w+14+2w
=6w+14
Please mark as brainliest if it’s correct
What is the following product? Assume x greater-than-or-equal-to 0 (4 x StartRoot 5 x squared EndRoot 2 x squared StartRoot 6 EndRoot) squared.
Answers
The simplified expression for the product [tex](4x\sqrt{5x^2} )(2x^2\sqrt{6})[/tex] is:
[tex]8\sqrt{30} x^4[/tex]
The given expression is:
[tex](4x\sqrt{5x^2} )(2x^2\sqrt{6})[/tex]
Note that :
[tex]\sqrt{5x^2} = x\sqrt{5}[/tex]
The given expression can then be re-written as:
[tex](4x^2\sqrt{5} )(2x^2\sqrt{6})[/tex]
Multiplying like terms together:
[tex]8x^4\sqrt{30}[/tex]
Finally, the resulting expression can be written as:
[tex]8\sqrt{30} x^4[/tex]
Learn more here: https://brainly.com/question/25913492
Answer:
D 104 x Superscript 4 + 16 x Superscript 4 Baseline StartRoot 30 EndRoot
Step-by-step explanation:
Edg 2022
4 divided by 3220 pls help me due in 2 hours
Answers
Answer:
1/805
Step-by-step explanation:
4/3220 = 1/805 = 0.00124223....
The product of double the number four
!!!Translate the following sentence into a variable expression!!!
Answers
Solve please in need of help
Answers
Answer:
not clear and I don't know the answer
Find 85 - 72 x 4 ÷ 2 (13 - 11)
Answers
Answer:
13
Step-by-step explanation:
using BODMAS
13-11=2
SO 2*2=4
4/4=1
1*72=72
THEREFORE it is 85-72 =13
Write ratios in the form 1:n or n:1 6 : 14
Answers
Answer:
3/7 : 1 OR 1 : 7/3
Step-by-step explanation:
You can set up ratios as a fraction and solve for the missing piece. See images.
Answer:
1 : 2.33 aka 1: 2 1/3
Step-by-step explanation:
1) 6 : 14
2) divide both sides by 6
6/6=1
14/6=2 1/3
3) 1 : 2 1/3
Mr. Russo had 6.25 cubic feet of potting soil in his raised garden bed. Then he took a bag of soil and poured in enough to completely fill the 9-cubic-foot bed. Let x represent how much soil was in the bag. Which inequality describes the problem?
Answers
Answer:
b
Step-by-step explanation: