Please help!

[s3velasc]

Hello I am working on solving inverse functions in my algebra and trig. class and I am haveing major problems. Can you please help me with a few problems I am having?

1. f(x)=x^2+9, x is greater than or equal to 0 is one-to-one. Find its inverse, and state the domain and range or f and f^-1.

Another area we are working on is Exponential Functions:

2. Solve A. 5^1-2x=1/5
B. (e^4)^x times e^x^2= e^12

Please Help!!

 

[s3velasc]

Can anyone help on these ones? I am really stuck on them.

Thanks

 

[tkhunny]

1. f(x)=x^2+9, x is greater than or equal to 0 is one-to-one. Find its inverse, and state the domain and range or f and f^-1.

Swap and solve. It is important to consider Domain and Range on a function for which an inverse is desired. for x ≥ 0, we get f(x) ≥ 9

If y = x^2 + 9, then start from x = y^2 + 9 ==> y = sqrt(x-9) and f^-1(x) = sqrt(x-9) For x ≥ 9, we get f^-1(x) ≥ 0.

Note: If we had started with f(x), for x < 0, we would have gotten f^-1(x) = -sqrt(x-9)

Another area we are working on is Exponential Functions:

2. Solve A. 5^1-2x=1/5
B. (e^4)^x times e^x^2= e^12

These I just can't understand. Use parentheses to clarify meaning. Is 5^1-2x this 5^(1-2x) or this (5^1)-2x. The second one seems a little silly, but I'm going to insist that you calrify it.

 

[soroban]

Hello, s3velasc!

I think I understand #2 . . .

2A) 5^1-2x = 1/5
.

We have: .5^1-2x .= .5^-1

Equate exponents: .1 - 2x .= .-1

Solve for x: .x = 1

2B) (e⁴)^x(e^{x^2}) = e¹²
.

We have: .(e^4x)(e^x²) .= .e¹²

Combine: .e^4x+x² .= .e¹²

Equate exponents: .4x + x² .= .12

We have a quadratic: .x² + 4x - 12 .= .0

. . which factors: .(x - 2)(x + 6) .= .0

. . and has roots: .x = 2, -6