Range of a Function: Find h(x)

So sqrt(25 + (x-3)^2) will never return a negative number. Therefore, the range of h(x) is y >= 5.In summary, the range of the function h(x) = sqrt(25 + (x - 3)2) is y >= 5. This is because the radicand, 25 + (x-3)2, will always be greater than or equal to 25, and the square root of any number cannot be negative. Therefore, the minimum value of h(x) is 5, and the range is y >= 5.
  • #1
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Homework Statement



Find the range of h

Homework Equations



h(x) = sqrt(25 + (x - 3)2)

The Attempt at a Solution



I factored out the (x-3)2 and simplified to get

sqrt(x2 - 6x + 34)

I was trying to figure out the domain first, knowing that x2 - 6x >= -34 in order for the number inside the sqrt to be non-negative.. I don't know, maybe I'm missing something really basic here. Help? :)
 
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  • #2
Codester09 said:

Homework Statement



Find the range of h

Homework Equations



h(x) = sqrt(25 + (x - 3)2)

The Attempt at a Solution



I factored out the (x-3)2 and simplified to get

sqrt(x2 - 6x + 34)
Let's get the terminology straight. You expanded (x-3)2 ; it was already factored.
Codester09 said:
I was trying to figure out the domain first, knowing that x2 - 6x >= -34 in order for the number inside the sqrt to be non-negative.. I don't know, maybe I'm missing something really basic here. Help? :)
The best thing, IMO, was to leave the radicand in its given form, 25 + (x-3)2. Looking at that as its own function, what is the range of this function? That will tell you a lot about the range of h(x).
 
  • #3
Well the range of 25 + (x - 3)2 is y >= 25, right? So, the range of sqrt(25 + (x - 3)2 is y >= 5?

Yea, I think expanding the (x - 3)2 term messed me up. I was thinking that i was possible for the radicand to be a negative number.. ugh. Stupid mistake.

Thanks a lot for the help.
 
  • #4
Also, just to let you know...

x^2-6x has a minimum of -9. It will never be less than -34.
 

1. What is the range of a function?

The range of a function refers to the set of all possible output values or y-values that a function can produce.

2. How do you find the range of a function?

To find the range of a function, you can graph the function and observe the y-values or output values. Alternatively, you can also use algebraic methods such as substitution or factoring to determine the range.

3. What is the importance of finding the range of a function?

The range of a function is important because it helps to understand the behavior and limitations of a function. It can also help in solving real-world problems and making predictions.

4. Can the range of a function be infinite?

Yes, the range of a function can be infinite if the function continues indefinitely in one direction and never repeats a y-value. This is often the case for exponential and logarithmic functions.

5. How can you determine if a function has a finite or infinite range?

If a function is given in equation form, you can look at the degree of the polynomial or the exponent of the function to determine if the range is finite or infinite. If the degree or exponent is odd, the range is typically infinite. If it is even, the range is usually finite. If the function is given in graph form, you can observe the behavior of the graph to determine the range.

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