Domain, Range & Absolute Value Functions Explained

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Let's dive into some cool math problems, guys! We're going to break down how to find the domain and range of functions, and then we'll tackle expressing functions without those pesky absolute value signs. Ready? Let's get started!

1. Finding the Domain and Range

Okay, so the domain is like the VIP list for a function – it's all the possible input values (usually x) that the function can handle without throwing a tantrum (like dividing by zero or taking the square root of a negative number). The range, on the other hand, is the result – it's all the possible output values (usually y or f(x)) that the function can spit out.

a) f(x)=2x+3{f(x) = \sqrt{2x+3}}

Let's start with the first function, f(x)=2x+3{f(x) = \sqrt{2x+3}}. This involves a square root, which means we need to be careful about what's inside. Remember, we can't take the square root of a negative number (at least, not in the world of real numbers!). So, we need to make sure that the expression inside the square root, 2x + 3, is greater than or equal to zero.

  • Finding the Domain:

    We set up the inequality:

    2x + 3 β‰₯ 0

    Now, let's solve for x:

    2x β‰₯ -3

    x β‰₯ -3/2

    So, the domain is all x values greater than or equal to -3/2. We can write this in interval notation as [-3/2, ∞). That fancy bracket means we include -3/2 in the domain.

  • Finding the Range:

    Now, what about the range? Well, the square root function always gives us non-negative results (zero or positive). The smallest value we can get from the square root is 0 (when 2x + 3 = 0). As x gets bigger, the value inside the square root gets bigger, and so does the square root itself. There's no upper limit!

    So, the range is all values greater than or equal to 0. In interval notation, that's [0, ∞).

b) f(x)=βˆ’625βˆ’y4{f(x)=-\sqrt{625-y^4}}

Next up, we've got f(x)=βˆ’625βˆ’y4{f(x)=-\sqrt{625-y^4}}. Notice that the variable inside is y this time, but the process is the same. We've still got a square root, and we've also got a negative sign hanging out front, which will affect our range.

  • Finding the Domain:

    Again, we need to make sure what's inside the square root is non-negative:

    625 - y⁴ β‰₯ 0

    This looks a bit trickier, but we can rearrange it:

    y⁴ ≀ 625

    Now, we need to think about what values of y would satisfy this. If we take the fourth root of both sides (remembering both positive and negative roots!), we get:

    -5 ≀ y ≀ 5

    So, the domain is all y values between -5 and 5, inclusive. In interval notation, that's [-5, 5].

  • Finding the Range:

    Here's where that negative sign comes into play. Without the negative sign, the square root would give us values from 0 upwards. But the negative sign flips that, so we'll be getting values from 0 downwards.

    The smallest value inside the square root (625 - y⁴) is 0 (when y = 5 or y = -5). The largest value is 625 (when y = 0). So, the square root part will give us values between √0 = 0 and √625 = 25. But because of the negative sign, our range will be between -25 and 0. In interval notation, that's [-25, 0].

2. Expressing Functions Without Absolute Values

Alright, let's talk about absolute values. Absolute value is like a distance meter – it tells us how far a number is from zero, regardless of direction. So, |5| is 5, and |-5| is also 5. When we have absolute values in a function, we need to think about how the expression inside the absolute value changes its sign.

We have the function f(x)=∣x∣+∣3x+1∣{f(x) = |x|+|3x+1|}. To get rid of the absolute values, we need to consider different cases based on when the expressions inside the absolute values are positive or negative.

  • Case 1: x < -1/3

    If x is less than -1/3, then x is negative, so |x| becomes -x. Also, 3x + 1 will be negative (since 3*(-1/3) + 1 = 0), so |3x + 1| becomes -(3x + 1).

    So, in this case, our function becomes:

    f(x) = -x - (3x + 1) = -x - 3x - 1 = -4x - 1

  • Case 2: -1/3 ≀ x < 0

    If x is between -1/3 (inclusive) and 0, then 3x + 1 will be non-negative (zero or positive), so |3x + 1| becomes 3x + 1. But x is still negative, so |x| becomes -x.

    So, in this case, our function becomes:

    f(x) = -x + (3x + 1) = -x + 3x + 1 = 2x + 1

  • Case 3: x β‰₯ 0

    If x is greater than or equal to 0, then both x and 3x + 1 are non-negative. So, |x| becomes x, and |3x + 1| becomes 3x + 1.

    So, in this case, our function becomes:

    f(x) = x + (3x + 1) = x + 3x + 1 = 4x + 1

    Putting it all together, we can write the function without absolute values as a piecewise function:

    f(x)={βˆ’4xβˆ’1,x<βˆ’1/32x+1,βˆ’1/3≀x<04x+1,xβ‰₯0{f(x) = \begin{cases} -4x - 1, & x < -1/3 \\ 2x + 1, & -1/3 \le x < 0 \\ 4x + 1, & x \ge 0 \end{cases}}

3. Piecewise Functions

Okay, let's tackle piecewise functions. These are functions that are defined differently over different intervals of their domain. It's like having different rules for different parts of the game.

We're given the function:

f(x)={1x,x>32x,x≀3{f(x) = \begin{cases} \frac{1}{x}, & x>3 \\ 2x, & x \le 3\end{cases}}

This function has two pieces:

  • For x values greater than 3, we use the rule f(x) = 1/x.
  • For x values less than or equal to 3, we use the rule f(x) = 2x.

To understand this function, we can think about what happens at different x values. For example:

  • If x = 4 (which is greater than 3), then f(4) = 1/4.
  • If x = 3, then f(3) = 2 * 3 = 6 (we use the second rule because x is equal to 3).
  • If x = 0, then f(0) = 2 * 0 = 0.
  • If x = -1, then f(-1) = 2 * (-1) = -2.

To fully analyze a piecewise function, you might want to consider things like its domain, range, continuity (does the graph have any breaks or jumps?), and end behavior (what happens as x gets very large or very small?).

Key takeaways about piecewise functions:

  • A piecewise function is defined by multiple sub-functions, each applying to a certain interval of the main function's domain.
  • When evaluating a piecewise function, it's crucial to first determine which interval the input value falls into.
  • Piecewise functions can exhibit interesting behaviors, like discontinuities, and require careful analysis.

In Conclusion:

So, guys, we've covered a lot in this breakdown! We’ve gone through how to find the domain and range of functions, how to deal with absolute values, and how to work with piecewise functions. Remember, the key is to break down each problem into smaller, manageable steps. Don't be afraid to take it slow and think carefully about each step. You've got this!