Domain of g(x): x≥−1
Range of g(x): y≤−3
Let's break down the transformations applied to the function f(x)= √x to obtain g(x):
Reflection across the x-axis: This means the positive values of
f(x) become negative and vice versa. For f(x)= √x , this reflection will make it f(x)=− √x
Translation down 3 units: This implies subtracting 3 from the function, so
g(x)=− √x −3.
Translation left 1 unit: This means replacing x with x+1, so g(x)=− √x+1 −3.
Now, let's analyze the domain and range of g(x):
Domain of g(x):
For the square root function, the domain is x≥0 because the square root of a negative number is not a real number. The transformation g(x)=− √x+1 −3 will not affect the domain since adding 1 inside the square root will still keep it within the non-negative domain. Therefore, the domain of
g(x) remains x≥−1.
Range of g(x):
The range of the original square root function f(x)= √x is y≥0 (all non-negative real numbers). The negative sign in g(x)=− √x+1 −3 reflects this graph across the x-axis, making the output negative. Also, the transformation of subtracting 3 shifts the entire graph downwards by 3 units. Therefore, the range of g(x) becomes y≤−3.