Let f left-parenthesis x right-parenthesis equals Start Root x End Root and lower g left parenthesis x right parenthesis equals 3 x plus 2. Find the range of left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis.
(1 point)
Responses
left-bracket 0 comma infinity right-parenthesis
- image with description: left-bracket 0 comma infinity right-parenthesis - - no response given
left-bracket 2 comma infinity right-parenthesis
- image with description: left-bracket 2 comma infinity right-parenthesis - - no response given
left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma infinity right-parenthesis
- image with description: left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma infinity right-parenthesis - - no response given
left-parenthesis negative infinity comma negative Start Fraction two over 3 End Fraction right-parenthesis union left-parenthesis negative Start Fraction 2 over 3 End Fraction comma 0 right-parenthesis union left-parenthesis 0 comma infinity right-parenthesis
The range of f(g(x)) can be found by substituting g(x) into f(x) and determining the possible values of f(g(x)).
First, substitute g(x) = 3x + 2 into f(x):
f(g(x)) = f(3x + 2) = √(3x + 2)
To find the range, we need to determine the possible values of √(3x + 2).
Since the square root function is always non-negative, the range of √(3x + 2) is [0, infinity).
Therefore, the range of f(g(x)) is left-bracket 0, infinity)
To find the range of f(g(x)), we need to determine all possible values of f(g(x)) as x varies.
Given f(x) = √x and g(x) = 3x + 2, we can substitute g(x) into f(x) to get:
f(g(x)) = √(3x+2)
To find the range of f(g(x)), we need to find the values that the expression √(3x+2) can take.
Since the square root function (√x) is defined for non-negative values only, we need to find when the expression inside the square root (√(3x+2)) is non-negative.
To do this, we set 3x+2 ≥ 0 and solve for x:
3x + 2 ≥ 0
3x ≥ -2
x ≥ -2/3
So, the range of f(g(x)) is given by:
(x) ≥ -2/3
Therefore, the range of f(g(x)) is (-2/3, ∞).