f(4) = 4 + 1/4 = 16/4 + 1/4 = 17/4

g is a function of f, and f is a function of 3, so you must work inside out.

f(3) = 11

g(f(3)) = g(11) = 121 + 11 = 132

The algebraic expression for x, the capacity of the fishtank is:

$\displaystyle 0.45x+7=0.65x$

$\displaystyle 0.45x+7-0.45x=0.65x-0.45x$

$\displaystyle 0.2x=7$

$\displaystyle 0.2x\times\frac{1}{0.2}=7\times\frac{1}{0.2}$

$\displaystyle x=35\hspace{1 mm}L$

The capacity of the fishtank is 35 liters.

When we multiply a function by a constant, we multiply each value in the function by that constant. Thus, 2f(x) = 4x + 12. We then subtract g(x) from that function, making sure to distribute the negative sign throughout the function. Subtracting g(x) from 4x + 12 gives us 4x + 12 - (3x - 3) = 4x + 12 - 3x + 3 = x + 15. We then add 2 to x + 15, giving us our answer of x + 17.

This equation describes a parabola whose vertex is located at the point (4, 6). No matter how large or small the value of t gets, the smallest that f(t) can ever be is 6 because the parabola is concave up. To prove this to yourself you can plug in different values of t and see if you ever get anything smaller than 6.

To find $\displaystyle f(x+h)$ when $\displaystyle f(x)=x^{2}+3$, we substitute $\displaystyle (x+h)$ for $\displaystyle x$ in $\displaystyle f(x)$.

Thus, $\displaystyle f(x+h)=(x+h)^{2}+3$.

We expand $\displaystyle (x+h)^{2}$  to $\displaystyle x^{2}+xh+xh+h^{2}$.

We can combine like terms to get $\displaystyle x^{2}+2xh+h^{2}$.

We add 3 to this result to get our final answer.

If $\displaystyle \dpi{100} h(x)$ and $\displaystyle \dpi{100} k(x)$ are defined as inverse functions, then $\displaystyle \dpi{100} h(k(x))=k(h(x))=x$. Thus, according to the definition of inverse functions, $\displaystyle \dpi{100} f(x)$ and $\displaystyle \dpi{100} g(x)$ given in the problem must be inverse functions.

If we want to find the inverse of a function, the most straighforward method is usually replacing $\displaystyle \dpi{100} f(x)$ with $\displaystyle \dpi{100} y$, swapping $\displaystyle \dpi{100} y$ and $\displaystyle \dpi{100} x$, and then solving for $\displaystyle \dpi{100} y$.

We want to find the inverse of $\displaystyle f(x)=\frac{1}{x-3}$. First, we will replace $\displaystyle \dpi{100} f(x)$ with $\displaystyle \dpi{100} y$.

$\displaystyle y = \frac{1}{x-3}$

Next, we will swap $\displaystyle \dpi{100} x$ and $\displaystyle \dpi{100} y$.

$\displaystyle x = \frac{1}{y-3}$

Lastly, we will solve for $\displaystyle \dpi{100} y$. The equation that we obtain in terms of $\displaystyle \dpi{100} x$ will be in the inverse of $\displaystyle \dpi{100} f(x)$, which equals $\displaystyle \dpi{100} g(x)$.

We can treat $\displaystyle x = \frac{1}{y-3}$ as a proportion, $\displaystyle \frac{x}{1} = \frac{1}{y-3}$. This allows us to cross multiply and set the results equal to one another.

$\displaystyle x(y-3)= 1$

We want to get y by itself, so let's divide both sides by x.

$\displaystyle y-3=\frac{1}{x}$

Next, we will add 3 to both sides.

$\displaystyle y=\frac{1}{x}+3$

To combine the right side, we will need to rewrite 3 so that it has a denominator of $\displaystyle \dpi{100} x$.

$\displaystyle y=\frac{1}{x}+3 = \frac{1}{x}+\frac{3x}{x}=\frac{3x+1}{x}$

The answer is $\displaystyle \frac{3x+1}{x}$.

$\displaystyle \small f\left ( x\right )=5x - 2$

$\displaystyle \small f\left ( 4\right )=5\left ( 4\right )-2 = 18$

$\displaystyle \small f\left ( 2\right )=5\left ( 2\right )-2 = 8$

$\displaystyle \small 18-8 = 10$

First we substitute in 12 for x and set the equation up as 12-t=4. We then get t=8, and substitute that for t and get f(0.5*8), giving us f(4). Plugging 4 in for x, and using t=8 that we found before, gives us:

f(4) = 4 - 8 = -4

To solve, we can set the given function equal to six and solve for $\displaystyle x$.

$\displaystyle 6= (x+3)^{2}-3$

Add 3 to each side:

$\displaystyle 9=(x+3)^2$

Take the square root. Remember that the root can be positive OR negative:

$\displaystyle \pm3=x+3$

Subtract 3 from each side. It will be easiest to separate the equation into two parts:

$\displaystyle 3=x+3\ \text{or}\ -3=x+3$

$\displaystyle 0=x\ \text{or}\ -6=x$

Now we know that $\displaystyle x$ is equal to $\displaystyle 0$ or $\displaystyle -6$. Based on the available answer options, the correct choice must be $\displaystyle f(0)$.

You can also solve this question by checking each answer option separately; you should find the same final answer.

Substitute each of the answer choices to see which one makes the equation equal to 6.

$\displaystyle f(1)=(1+3)^{2}-3=4^2-3=16-3=13$

$\displaystyle f(-1) = (-1+3)^2-3 = 2^2-3 = 4-3 =1$ 

$\displaystyle f(-2) =(-2+3)^2-3 = 1^2 - 3 = 1-3= -2$

$\displaystyle f(-4)= (-4+3)^2 - 3 = (-1)^2 - 3 = 1-3 = - 2$

$\displaystyle f(0) = (0+3)^2 -3=9-3=6$

The answer to this question is $\displaystyle f(0)$. No other answer makes the equation equal 6.