Find g of and f∘g, if f:R→R and g:R→R are given by f(x)=cosx and g(x)=3x2. Show that g of = fog.
Get the complete, step-by-step math solution for: "Find g of and f g, if f: {R} → {R} and g: {R} → {R} are given by f(x)=cos x and g(x)=3 x^{2}. Show that g of ≠ fog.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the functions
We are given two functions, f(x)=cosx and g(x)=3x2. Both functions map real numbers to real numbers.
Step 2: Calculate g∘f
To find g∘f(x), we substitute f(x) into g(x). Since f(x)=cosx, we replace x in g(x) with cosx. This gives us 3(cosx)2, which simplifies to 3cos2x.
Step 3: Calculate f∘g
To find f∘g(x), we substitute g(x) into f(x). Since g(x)=3x2, we replace x in f(x) with 3x2. This results in cos(3x2).
Step 4: Compare g∘f and f∘g
By comparing the results, we see that g∘f(x)=3cos2x and f∘g(x)=cos(3x2). These two expressions are generally not equal for all values of x. For example, if x=0, g∘f(0)=3cos2(0)=3(1)2=3, while f∘g(0)=cos(3(0)2)=cos(0)=1. Since 3=1, we have shown that g∘f=f∘g.