Reflections and Symmetry

Question

Give the formula for each function. a)f(-x) and b)-f(x)

1) f(x)=1/x.....I know you have to plug in ^ into x...and I got a) 1/-x and b) 1/-(x).....is that right ?

2) f(x)= 3x³/(x²-1)...and for this one I got a) -3x³/x-1 and b) -3x³/-x²-1.....is that one right ?

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The graph of P=g(t) contains the point (-1,-5)

3) if the graph has even symmetry, which other point must lie on the graph? For this one I got (1,5) ???

4) what point must lie on the graph -g(t)???< im not sure how to do this one.

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5) The range of Q(x) is -2≤Q(x)≤12. What is the range of -Q(x)?

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The function Q(t) has a domain t>0 and range f -4≤Q(t)≤7. Give the domain and range for the transformation of Q(t).

6) y=Q(-t)

7) y=-Q(t)

8) y=-Q(-t)

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Last one :)

9) if it is 3^(-x) would you keep it as 3^(-x) or would it change to 3^-x ?

THANK YOU SOOOOOOO MUCH FOR HELPING ME !!!!!!!!!!!!! :)

Answer 1

Reflections and Symmetry

Give the formula for each function.
a) f(-x)
b) -f(x)

1) f(x) = 1/x
a) f(-x) = 1/(-x) = –1/x
b) -f(x) = –1/x
I got a) 1/-x and b) 1/-(x).....is that right ?
Yes; but enclose the entire denominator in parentheses.

2) f(x) = 3x^3/(x^2-1)
I got a) -3x^3/x-1 and b) -3x^3/-x^2-1.....is that one right ?
No. You must use parentheses; like this:
a) f(-x) = 3(-x)^3/((-x)^2-1) = -3x^3/(x^2-1)
b) -f(x) = -3x^3/(x^2-1)
(-x)^2 = (-x)(-x) = +x^2
(-x)^3 = (-x)(-x)(-x) = -x^3

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The graph of P = g(t) contains the point (-1,-5)
3) if the graph has even symmetry, which other point must lie on the graph?

For this one I got (1,5) ???
No. Even symmetry means the graph can be reflected over the y-axis onto itself. E.g., y = x^2 has even symmetry and its Axis of Symmetry is the y-axis.
So, (-1,-5) reflected over the y-axis is (1,-5). Graph the points to see it.

4) what point must lie on the graph -g(t)
??? I’m not sure how to do this one.

We usually graph a function’s value along the y axis and write y = g(t). So -g(t) is g(t) reflected over the x-axis; i.e., all the y-values of the points become the opposites. Since (-1,-5) is on g(t), then (-1,5) is on -g(t).

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5) The range of Q(x) is -2 ≤ Q(x) ≤ 12. What is the range of -Q(x)?

Here’s a crude drawing of the range of Q(x) on the y-axis:

^ y
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* 12
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|| 0
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* –2
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Since -Q(x) is Q(x) reflected over the x-axis, what do you think you’d do with the drawing?

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The function Q(t) has a
domain t > 0 and range of -4 ≤ Q(t) ≤ 7.

Give the domain and range for the transformation of Q(t).
6) y=Q(-t)
7) y=-Q(t)
8) y=-Q(-t)

Here are the rules:
To reflect over the y-axis change t to -t.
To reflect over the x-axis change y to -y.
To reflect over the origin do both.

Try it.

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Last one :)

9) if it is 3^(-x) would you keep it as 3^(-x) or would it change to 3^-x ?

Please don’t use superscripts for exponents because they don’t copy as superscripts. Use ^ for exponents. And if there is more than one character in the exponent expression enclose it in parentheses.

So for all three of your examples, use 3^(-x).