Let f(x)=√x. Find g(x), the function that is f(x) shifted up 2 units and left 5 units.
g(x) = f(x+5)+2 = √(x+5)+2
Well, if f(x) is √x, then g(x), the function shifted up 2 units, would be g(x) = √x + 2. And if we want to shift it left 5 units, we need to replace x with (x + 5). So, g(x) = √(x + 5) + 2. Or as I like to call it, the "sqrt(x) + loot" function!
To find the function g(x) that is f(x) shifted up 2 units and left 5 units, we can apply the following transformations:
1. Upward shift: To shift the function f(x) up 2 units, we add a positive constant to the function. Therefore, we have g(x) = f(x) + 2.
2. Left shift: To shift the function g(x) left 5 units, we subtract a positive constant from x. Therefore, we have g(x) = f(x - 5).
Since f(x) = √x, we can substitute this into the above equation to find the final expression for g(x):
g(x) = √(x - 5) + 2.
Thus, g(x) is the function obtained by shifting f(x) up 2 units and left 5 units and is given by g(x) = √(x - 5) + 2.
To find the function g(x) that is a transformation of f(x), we will apply two transformations: shifting up 2 units and shifting left 5 units.
1. Shifting up 2 units:
To shift the graph of f(x) up 2 units, we can add 2 to the original function f(x). Therefore, g(x) can be defined as f(x) + 2.
2. Shifting left 5 units:
To shift the graph of g(x) left 5 units, we can replace the variable x with (x + 5). Hence, g(x) can be expressed as f(x + 5).
Combining these transformations, we can write the function g(x) as g(x) = f(x + 5) + 2.
Now let's simplify g(x) further using the given function f(x) = √x:
g(x) = f(x + 5) + 2
= √(x + 5) + 2.
Therefore, the function g(x) that represents f(x) shifted up 2 units and left 5 units is g(x) = √(x + 5) + 2.