Question 20·SAT Desmos Regression Practice·Advanced Math

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The function $f$ is defined by

f(x)=\frac{a}{x-1}+2

where $a$ is a constant. If $f(4)=5$ , which choice is the value of $f(-1)$ ?

A parameter is a constant in a function rule, and a known input-output pair can fix its value. Substitute the known input into the rule and set it equal to the known output. A Desmos regression finds the constant; then define the function and evaluate the requested input. At a negative input, watch the sign of the denominator.

Hints

Hint 1
In function notation, $f(4)=5$ means the input $4$ gives the output $5$ . To find the unknown constant, replace $x$ with $4$ in the rule and set the result equal to $5$ .
Hint 2
Evaluating a function means putting an input into its rule. Once you know $a$ , replace $x$ with the entire negative input $-1$ . What does the denominator $x-1$ become?

Step-by-step

Find the constant, then evaluate
Step 1
Turn the known output into an equation
$f(4)=5$ means putting in $4$ gives an output of $5$ . Set the rule at $4$ equal to that known output:
$\frac{a}{4-1}+2=5.$
Find the constant from the known output before evaluating the new input.
Step 2
Find the constant in Desmos
A regression asks Desmos to find an unknown value that fits a fact. Type $\frac{a}{4-1}+2\sim5$ ; the tilde $\sim$ tells Desmos to find the constant that makes the equation true. Under PARAMETERS, it shows $a=9$ .
Step 3
Evaluate at the requested input
Type $f(x)=\frac{a}{x-1}+2$ , then $f(-1)$ . Desmos reuses $a=9$ . At this input, the denominator is $(-1)-1=-2$ , so keep the negative sign. Desmos shows $-2.5$ ; click its fraction button to see $-\frac{5}{2}$ . So $f(-1)=-\frac{5}{2}$ . Choice A.

Question 20·SAT Desmos Regression Practice·Advanced Math

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