SAT Math - How to find f(x)

If f(x) = _x_2 – 5 for all values x and f(a) = 4, what is one possible value of a?

Explanation

Using the defined function, f(a) will produce the same result when substituted for x:

f(a) = _a_2 – 5

Setting this equal to 4, you can solve for a:

_a_2 – 5 = 4

_a_2 = 9

a = –3 or 3

An outpost has the supplies to last 2 people for 14 days. How many days will the supplies last for 7 people?

$\dpi{100} \small 4$

$\dpi{100} \small 7$

$\dpi{100} \small 5$

$\dpi{100} \small 10$

$\dpi{100} \small 9$

Explanation

Supplies are used at the rate of $\dpi{100} \small \frac{Supplies}{Days\times People}$ .

Since the total amount of supplies is the same in either case, $\dpi{100} \small \frac{1}{14\times 2}=\frac{1}{7\times \ (&hash;\ of\ days)}$ .

Solve for days to find that the supplies will last for 4 days.

Explanation

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.

f(x) = 4x + 2

g(x) = 3x - 1

The two equations above define the functions f(x) = g(x). If f(d) = 2g(d) for some value of d, then what is the value of d?

-2

1/2

3/2

Explanation

f(x) = 4x + 2

g(x) = 3x - 1

We have f(d) = 2g(d). We multiply each value in g(d) by 2.

4d + 2 = 2(3d - 1) (Distribute the 2 in the parentheses by multiplying each value in them by 2.)

4d + 2 = 6d - 2 (Add 2 to both sides.)

4d + 4 = 6d (Subtract 4d from both sides.)

4 = 2d (Divide both sides by 2.)

2 = d

We can plug that back in to double check.

4(2) + 2 = 6(2) - 2

8 + 2 = 12 - 2

10 = 10

Let $f(x)$ and $g(x)$ be functions such that $f(x)=\frac{1}{x-3}$ , and $f(g(x))=g(f(x))=x$ . Which of the following is equal to $g(x)$ ?

$\frac{3x+1}{x}$

$\frac{1}{x+3}$

$\frac{x}{x-3}$

$\frac{4}{x}$

$\frac{x}{x+3}$

Explanation

If $\dpi{100} h(x)$ and $\dpi{100} k(x)$ are defined as inverse functions, then $\dpi{100} h(k(x))=k(h(x))=x$ . Thus, according to the definition of inverse functions, $\dpi{100} f(x)$ and $\dpi{100} g(x)$ 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 $\dpi{100} f(x)$ with $\dpi{100} y$ , swapping $\dpi{100} y$ and $\dpi{100} x$ , and then solving for $\dpi{100} y$ .

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

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

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

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

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

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

$x(y-3)= 1$

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

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

Next, we will add 3 to both sides.

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

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

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

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

The cost of a cell phone plan is $40 for the first 100 minutes of calls, and then 5 cents for each minute after. If the variable x is equal to the number of minutes used for calls in a month on that cell phone plan, what is the equation f(x) for the cost, in dollars, of the cell phone plan for calls during that month?

f(x) = 40 + 5x

f(x) = 40 + 0.5x

f(x) = 40 + 0.05x

f(x) = 40 + 0.5(x - 100)

f(x) = 40 + 0.05(x - 100)

Explanation

40 dollars is the constant cost of the cell phone plan, regardless of minute usage for calls. We then add 5 cents, or 0.05 dollars, for every minute of calls over 100. Thus, we do not multiply 0.05 by x, but rather by (x - 100), since the 5 cent charge only applies to minutes used that are over the 100-minute barrier. For example, if you used 101 minutes for calls during the month, you would only pay the 5 cents for that 101st minute, making your cost for calls $40.05. Thus, the answer is 40 + 0.05(x - 100).

Let the function f be defined by f(x)=x-t. If f(12)=4, what is the value of f(0.5*t)?

Explanation

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

The function f, where f(x) = x2 + 6x + 8, is related to function g, where g(x) = 5 f(x-2). What is g(3)?

125

150

175

Explanation

Doing things in an orderly way is a friend to the test-taker.

g(3) = 5 f(3-2)

= 5 f(1)

= 5 \[ 12 + 6**∙**1 + 8\]

= 5 \[ 1 + 6 + 8\]

= 5 \[ 15\]

= 75

The rate of a gym membership costs p dollars the first month and m dollars per month every month thereafter. Which of the following represents the total cost of the gym membership for n months, if n is a positive integer?

p+mn

p+m(n+1)

p+m(n-1)

pn+m(n-1)

Explanation

The one-time first-month cost is p, and the monthly cost is m, which gets multipled by every month but the first (of which there are n -1). The total cost is the first-month cost of p, plus the monthly cost for (i.e. times) n -1 months, which makes the total cost equal to p + m (n -1).

The point is one of the points of intersection of the graphs of and . Given that and , find the value of .