Learn Intro to Composing Functions & fg x, practice Composition of Functions Example.

Let us say, we have two functions 'f' and 'g' ,then we can make new functions from these by suitable combinations. Four such combinations are

(1) \(f + g \) (3) \(f g \)

(2) \(f - g \) (4) \(\dfrac {f}{g }\)

The definition is similar to the way we define addition, subtraction, division & multiplication in real numbers.

So

\((f + g) (x) = f(x) + g(x) \) \( fg(x) = f(x)\; g(x)\)

\((f – g) (x) = f(x) – g(x)\) \(\Big(\dfrac {f}{g}\Big)(x) = \Large \frac {f(x)}{g(x)}\)

A \(x^2 + 8x\)

B \(7x^2 + 9x - 10\)

C \(x^2 + 6x - 1\)

D \(x^2 + x\)

A \(\dfrac {5}{8}\)

B \(\dfrac {-8}{5}\)

C \(\dfrac {7}{3}\)

D \(\dfrac {-3}{7}\)

- The product function of two function \(f\) and \(g\) is defined as \((fg)\,(x)=f(x)\,g(x)\)
- If this product function is to be found at particular value of \(x\) say \(u=\alpha\), then \((fg)\,(\alpha)=f(\alpha)\,g(\alpha)\). So find \(f(\alpha)\) and \(g(\alpha)\) separately and obtain the product.

A \(\sqrt {3x+5}\,(x^2+2)\)

B \((2x^2+7)\;\sqrt {2+x}\)

C \(\dfrac {3+x^2}{x}\)

D \(\dfrac {x+7}{9}\)

- The division function of two \( f \) function and \( g\) is defined as

\(h(x)=\dfrac {f}{g}(x)=\dfrac {f(x)}{g(x)}\rightarrow\) by definition

- If this division function is to be found at a particular value of \(x\) say \(x=\alpha\) ,then

\((f \,/g)(\alpha)=f(\alpha)/g(\alpha)\)

so find \( f(\alpha)\) and \(g(\alpha)\) separately and obtain the division.

A \(\dfrac {5x+3}{x+7}\)

B \(\dfrac {2x^2+3x+1}{x+2}\)

C \(\dfrac {x-2}{x^2+1}\)

D \(\dfrac {\sqrt {x+3}}{x+6}\)