The function is defined by If , what is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 167 | Free SAT Prep | Aniko

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Question 167·Medium·Nonlinear Functions

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The function $f$ is defined by

f(x)=7\, 2^{x-3}+5.

If $f(x)=61$ , what is the value of $x$ ?

For exponential equations like this on the SAT, isolate the exponential part first by undoing any added or multiplied numbers in reverse order (subtract, then divide). Once you have something like $2^{\text{expression}} = \text{number}$ , rewrite the number as a power of the same base (here, 2) and then set the exponents equal to form a simple linear equation. This method is usually faster and less error-prone than guessing from the choices, though you can always plug answer choices into the original function as a quick check if you are unsure.

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Write the equation to solve

Set $f(x)$ equal to 61 and write the full equation involving $2^{x-3}$ .

Get the exponential term alone

Try subtracting 5 from both sides first, then see what you can do to remove the 7 from in front of $2^{x-3}$ .

Recognize a power of 2

After isolating $2^{x-3}$ , you should get a simple number. Can you write that number as $2$ raised to some power?

Use equal bases

Once both sides are written as powers of 2, compare the exponents to form a simple linear equation for $x$ .

Desmos Guide

Graph the function and the target value

In Desmos, enter y = 7*2^(x-3) + 5 on one line and y = 61 on another line to create the function graph and a horizontal line at 61.

Find the intersection point

Use Desmos to tap or click on the point where the graph of $y = 7\cdot 2^{x-3} + 5$ intersects the line $y = 61$ . Read off the $x$ -coordinate of that intersection; that $x$ -value is the solution to the equation $f(x)=61$ .

Step-by-step Explanation

Set up the equation

Start by setting the function equal to 61:

7\cdot 2^{x-3} + 5 = 61

Isolate the exponential part

Subtract 5 from both sides to get the exponential term by itself:

7\cdot 2^{x-3} = 61 - 5

7\cdot 2^{x-3} = 56

Now divide both sides by 7:

2^{x-3} = \dfrac{56}{7} = 8

Rewrite 8 as a power of 2

Notice that 8 is a power of 2:

8 = 2^3

So the equation $2^{x-3} = 8$ becomes

2^{x-3} = 2^3

Equate exponents and solve for x

When the bases are the same (both 2), their exponents must be equal:

x - 3 = 3

Add 3 to both sides:

x = 6

So the value of $x$ is 6.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation