# Calculate ??( hofog)x if h(x) = 3x , g = 1/3x and f =2x+5

May 8, 2018

$\left(h \circ f \circ g\right) \left(x\right) = 2 x + 15$

#### Explanation:

$\left(h \circ f \circ g\right) \left(x\right)$ is known as a composite function. Here's how composite functions work:

let's say that $x = 1$. Your function $g \left(x\right) = \frac{1}{3} \left(x\right)$ now produces a $y$ output of $\frac{1}{3}$ , since $g \left(1\right) = \frac{1}{3} \left(1\right)$. Within a composite function, the y-value of one function becomes the x-value of the next, like so:

$g \left(1\right) = \frac{1}{3} \implies f \left(\frac{1}{3}\right) = \frac{17}{3} \implies h \left(\frac{17}{3}\right) = 17$

Therefore,
$\left(h \circ f \circ g\right) \left(1\right) = 17$

Based on this, to find the function for $\left(h \circ f \circ g\right) \left(x\right)$ (combined from right to left, by the way), simply replace $x$ in $f \left(x\right)$ with the function $g \left(x\right)$, and replace $x$ in $h \left(x\right)$ with the function of $\left(f \circ g\right) \left(x\right)$, to get $\left(h \circ f \circ g\right) \left(x\right)$.

This, simplified, is equal to $2 x + 15$, and therefore $\left(h \circ f \circ g\right) \left(1\right) = 2 \left(1\right) + 15 = 17$

Hope that helps!