Combining and Composing Functions: Additional Practice (Set 1)

The voltage feeds into the display, so substitute $v(t) = 2t + 1$ into $r$ and distribute.

$r(v(t)) = 3(2t + 1) = 6t + 3$

The bill feeds into the reimbursement, so halve the whole cost rule.

$r(c(m)) = \frac{0.1m + 20}{2} = 0.05m + 10$

Both terms are divided by $2$.

Here $(f \circ g)(x) = \dfrac{1}{x - 3}$, and the outer function cannot take $0$, so exclude where $g(x) = 0$.

$x - 3 = 0 \;\Longrightarrow\; x = 3$

The domain is all numbers except $3$.

The inner function is $g$, so substitute $x^2$ into $f$.

$f(g(x)) = 2(x^2) + 1 = 2x^2 + 1$

Here $f$ acts last, doubling and then adding $1$.

Now $f$ is the inner function, so square the whole rule $2x + 1$.

$g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1$

This differs from $f(g(x)) = 2x^2 + 1$, since order matters.

Substitute the whole rule $g(x) = x + 3$ into $f$, distribute the $2$, then combine constants.

$(f \circ g)(x) = 2(x + 3) - 1 = 2x + 6 - 1 = 2x + 5$

The gross pay feeds into the after-tax rule, so keep $80$ percent of $15h$.

$n(w(h)) = 0.8(15h) = 12h$

Run $g$ on $-1$ first, then square. Here $g(-1) = -1 + 3 = 2$.

$f(2) = 2^2 = 4$

So $f(g(-1)) = 4$.

Run $g$ on $2$ first, then square. Here $g(2) = 2(2) - 1 = 3$.

$f(3) = 3^2 = 9$

So $f(g(2)) = 9$.

Discount first, then add shipping to the discounted price, so evaluate $s(d(40))$.

$s(d(40)) = 0.75(40) + 5 = 30 + 5 = 35$

Adding shipping before discounting would instead give $0.75(45) = 33.75$ dollars.

The inner function is $f$, so add $2$ to the whole rule $4x - 1$.

$g(f(x)) = (4x - 1) + 2 = 4x + 1$

The height feeds into the volume rule, so substitute $h(t) = 2t$ into $V$.

$V(h(t)) = 5(2t) = 10t$