4. State the Domain and Range of the following:
a) $f(x)=2(x-1)^{\wedge} 2-4$
b) $f(x)=5(x+1)^{\wedge} 2+1$
Final Answer: \(\boxed{\text{b) Domain: }(-\infty, \infty), \text{ Range: }[1, \infty)}\)
Step 1 :Given functions: \(f(x)=2(x-1)^2-4\) and \(f(x)=5(x+1)^2+1\)
Step 2 :Both functions are quadratic functions in the form of \(f(x)=a(x-h)^2+k\)
Step 3 :The domain of a quadratic function is all real numbers
Step 4 :To find the range, determine if the parabola opens upwards or downwards and find the vertex
Step 5 :For \(f(x)=2(x-1)^2-4\), the parabola opens upwards and the vertex is \((1, -4)\)
Step 6 :For \(f(x)=5(x+1)^2+1\), the parabola opens upwards and the vertex is \((-1, 1)\)
Step 7 :Final Answer: \(\boxed{\text{a) Domain: }(-\infty, \infty), \text{ Range: }[-4, \infty)}\)
Step 8 :Final Answer: \(\boxed{\text{b) Domain: }(-\infty, \infty), \text{ Range: }[1, \infty)}\)