Question 358·Hard·Math

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Let $f(x) = a\,b^x$ , where $a \neq 0$ and $b > 0$ , $b \neq 1$ . Define $g(x) = f(x+2) - 4f(x+1) + 4f(x)$ . For which value of $b$ is $g(x) = 0$ for all real $x$ ?

For an exponential function $f(x)=ab^x$ , increasing $x$ by $1$ multiplies the output by $b$ . When shifted copies of the same function are added, rewrite each as a multiple of $f(x)$ . Factor out the original exponential before solving for its base. Then use the given conditions to decide which factor can equal zero, rather than matching $b$ to one coefficient.

Hints

Hint 1
A one-step shift in the exponent multiplies an exponential output by its base: $f(x+1)=bf(x)$ . A two-step shift multiplies it by $b^2$ . What factor will all three terms of $g(x)$ share?
Hint 2
Once you factor out $f(x)$ , look at all three terms of the remaining quadratic. A perfect-square trinomial has the form $u^2-2uv+v^2$ . What squared expression matches the quadratic?
Hint 3
Because $b>0$ , $b^x$ is positive for every real $x$ . Also, $a\ne0$ , so $f(x)$ cannot be zero. In the product for $g(x)$ , which factor must be zero?

Step-by-step

Factor out the exponential
Step 1
Rewrite the one-step shift
From $f(x)=ab^x$ , adding $1$ to the exponent multiplies the output by the base $b$ . Write the one-step shift:
$f(x+1)=bf(x).$
Step 2
Rewrite the two-step shift
Apply the same shift rule twice for the two-step shift:
$f(x+2)=b^2f(x).$
Step 3
Substitute the shifted outputs
Substitute those shifted outputs into the given expression for $g(x)$ :
$g(x)=b^2f(x)-4bf(x)+4f(x).$
Step 4
Pull out the common factor
Factor out the common factor $f(x)$ :
$g(x)=f(x)(b^2-4b+4).$
Step 5
Recognize the square
Squaring $b-2$ gives $b^2-4b+4$ , a perfect-square trinomial. Factor all three terms together:
$g(x)=f(x)(b-2)^2.$
Step 6
Use the nonzero condition
Since $b>0$ , $b^x>0$ for every real $x$ . With $a\ne0$ , $f(x)=ab^x$ is never zero. Divide $g(x)=0$ by $f(x)$ :
$(b-2)^2=0.$
Step 7
Set the expression inside the square to zero
A square is zero only when what's inside it is zero:
$b-2=0.$
Step 8
Solve for the base
Type $b_1-2\sim0$ in Desmos. The subscript makes $b_1$ an unknown for the regression to find, and $\sim$ tells Desmos to match the two sides. Under PARAMETERS, it shows $b_1=2$ . This base is positive and not $1$ , and it makes $g(x)=0$ for every real $x$ . Choice C.

Question 358·Hard·Math

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